natural logarithm and natural exponentialapilking/math10560/lectures/lecture... · 2020-01-13 ·...
TRANSCRIPT
![Page 1: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/1.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Definition and properties of ln(x).
We define a new function
ln x =
Z x
1
1
tdt, x > 0.
This function is called the natural logarithm. We derive a number of propertiesof this new function:
I Domain = (0,∞)
I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.
I d(ln x)dx
= 1x
I The graph of y = ln x is increasing, continuous and concave down on theinterval (0,∞).
I The function f (x) = ln x is a one-to-one function
I Since f (x) = ln x is a one-to-one function, there is a unique number, e,with the property that
ln e = 1.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 2: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/2.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Definition and properties of ln(x).
We define a new function
ln x =
Z x
1
1
tdt, x > 0.
This function is called the natural logarithm. We derive a number of propertiesof this new function:
I Domain = (0,∞)
I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.
I d(ln x)dx
= 1x
I The graph of y = ln x is increasing, continuous and concave down on theinterval (0,∞).
I The function f (x) = ln x is a one-to-one function
I Since f (x) = ln x is a one-to-one function, there is a unique number, e,with the property that
ln e = 1.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 3: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/3.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Definition and properties of ln(x).
We define a new function
ln x =
Z x
1
1
tdt, x > 0.
This function is called the natural logarithm. We derive a number of propertiesof this new function:
I Domain = (0,∞)
I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.
I d(ln x)dx
= 1x
I The graph of y = ln x is increasing, continuous and concave down on theinterval (0,∞).
I The function f (x) = ln x is a one-to-one function
I Since f (x) = ln x is a one-to-one function, there is a unique number, e,with the property that
ln e = 1.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 4: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/4.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Definition and properties of ln(x).
We define a new function
ln x =
Z x
1
1
tdt, x > 0.
This function is called the natural logarithm. We derive a number of propertiesof this new function:
I Domain = (0,∞)
I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.
I d(ln x)dx
= 1x
I The graph of y = ln x is increasing, continuous and concave down on theinterval (0,∞).
I The function f (x) = ln x is a one-to-one function
I Since f (x) = ln x is a one-to-one function, there is a unique number, e,with the property that
ln e = 1.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 5: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/5.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Definition and properties of ln(x).
We define a new function
ln x =
Z x
1
1
tdt, x > 0.
This function is called the natural logarithm. We derive a number of propertiesof this new function:
I Domain = (0,∞)
I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.
I d(ln x)dx
= 1x
I The graph of y = ln x is increasing, continuous and concave down on theinterval (0,∞).
I The function f (x) = ln x is a one-to-one function
I Since f (x) = ln x is a one-to-one function, there is a unique number, e,with the property that
ln e = 1.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 6: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/6.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Definition and properties of ln(x).
We define a new function
ln x =
Z x
1
1
tdt, x > 0.
This function is called the natural logarithm. We derive a number of propertiesof this new function:
I Domain = (0,∞)
I ln x > 0 if x > 1, ln x = 0 if x = 1, ln x < 0 if x < 1.
I d(ln x)dx
= 1x
I The graph of y = ln x is increasing, continuous and concave down on theinterval (0,∞).
I The function f (x) = ln x is a one-to-one function
I Since f (x) = ln x is a one-to-one function, there is a unique number, e,with the property that
ln e = 1.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 7: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/7.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Graph of ln(x).
Using the information derived above, we can sketch a graph of the naturallogarithm
e1 2 3 4 5
-0.5
0.5
1.0
1.5
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 8: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/8.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Graph of ln(x).
Using the information derived above, we can sketch a graph of the naturallogarithm
e1 2 3 4 5
-0.5
0.5
1.0
1.5
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 9: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/9.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Rules of Logarithms
We also derived the following algebraic properties of our new function bycomparing derivatives. We can use these algebraic rules to simplify the naturallogarithm of products and quotients:
III ln 1 = 0
I ln(ab) = ln a + ln b
I ln ar = r ln a
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 10: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/10.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Rules of Logarithms
We also derived the following algebraic properties of our new function bycomparing derivatives. We can use these algebraic rules to simplify the naturallogarithm of products and quotients:
I ln 1 = 0
I ln(ab) = ln a + ln b
I ln ar = r ln a
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 11: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/11.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Rules of Logarithms
We also derived the following algebraic properties of our new function bycomparing derivatives. We can use these algebraic rules to simplify the naturallogarithm of products and quotients:
I ln 1 = 0
I ln(ab) = ln a + ln b
I ln ar = r ln a
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 12: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/12.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Rules of Logarithms
We also derived the following algebraic properties of our new function bycomparing derivatives. We can use these algebraic rules to simplify the naturallogarithm of products and quotients:
I ln 1 = 0
I ln(ab) = ln a + ln b
I ln ar = r ln a
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 13: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/13.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Example using definition of e and rule 3
Example EvaluateR e2
11tdt
I From the definition of ln(x), we haveZ e2
1
1
tdt = ln(e2)
I
= 2 ln e = 2.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 14: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/14.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Example using definition of e and rule 3
Example EvaluateR e2
11tdt
I From the definition of ln(x), we haveZ e2
1
1
tdt = ln(e2)
I
= 2 ln e = 2.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 15: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/15.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Example using definition of e and rule 3
Example EvaluateR e2
11tdt
I From the definition of ln(x), we haveZ e2
1
1
tdt = ln(e2)
I
= 2 ln e = 2.
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 16: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/16.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Limits at ∞ and 0.
We can use the rules of logarithms given above to derive the followinginformation about limits.
limx→∞
ln x =∞, limx→0
ln x = −∞.
(see notes for a proof)Example Find the limit limx→∞ ln( 1
x2+1).
I As x →∞, we have 1x2+1
→ 0
I Therfore as x →∞, ln( 1x2+1
)→ −∞ [= limu→0 ln(u)]
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 17: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/17.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Limits at ∞ and 0.
We can use the rules of logarithms given above to derive the followinginformation about limits.
limx→∞
ln x =∞, limx→0
ln x = −∞.
(see notes for a proof)Example Find the limit limx→∞ ln( 1
x2+1).
I As x →∞, we have 1x2+1
→ 0
I Therfore as x →∞, ln( 1x2+1
)→ −∞ [= limu→0 ln(u)]
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 18: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/18.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Limits at ∞ and 0.
We can use the rules of logarithms given above to derive the followinginformation about limits.
limx→∞
ln x =∞, limx→0
ln x = −∞.
(see notes for a proof)Example Find the limit limx→∞ ln( 1
x2+1).
I As x →∞, we have 1x2+1
→ 0
I Therfore as x →∞, ln( 1x2+1
)→ −∞ [= limu→0 ln(u)]
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 19: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/19.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
ln |x |We can extend the applications of the natural logarithm function by composingit with the absolute value function. We have :
ln |x | =
ln x x > 0
ln(−x) x < 0
This is an even function with graph
-20 -10 10 20
-1
1
2
3
We have ln|x | is also an antiderivative of 1/x with a larger domain than ln(x).
d
dx(ln |x |) =
1
xand
Z1
xdx = ln |x |+ C
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 20: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/20.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Chain Rule
d
dx(ln |x |) =
1
xand
d
dx(ln |g(x)|) =
g ′(x)
g(x)
Example Differentiate ln | sin x |.
I Using the chain rule, we have
d
dxln | sin x | =
1
sin x
d
dxsin x
I
=cos x
sin x
Example Differentiate ln | 3√
x − 1|.
I We can simplify this to finding ddx
“13
ln |x − 1|”
, since
ln | 3√
x − 1| = ln |x − 1|1/3
Id
dx
1
3ln |x − 1| =
1
3
1
(x − 1)
d
dx(x − 1) =
1
3(x − 1)
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 21: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/21.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Chain Rule
d
dx(ln |x |) =
1
xand
d
dx(ln |g(x)|) =
g ′(x)
g(x)
Example Differentiate ln | sin x |.I Using the chain rule, we have
d
dxln | sin x | =
1
sin x
d
dxsin x
I
=cos x
sin x
Example Differentiate ln | 3√
x − 1|.
I We can simplify this to finding ddx
“13
ln |x − 1|”
, since
ln | 3√
x − 1| = ln |x − 1|1/3
Id
dx
1
3ln |x − 1| =
1
3
1
(x − 1)
d
dx(x − 1) =
1
3(x − 1)
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 22: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/22.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Chain Rule
d
dx(ln |x |) =
1
xand
d
dx(ln |g(x)|) =
g ′(x)
g(x)
Example Differentiate ln | sin x |.I Using the chain rule, we have
d
dxln | sin x | =
1
sin x
d
dxsin x
I
=cos x
sin x
Example Differentiate ln | 3√
x − 1|.
I We can simplify this to finding ddx
“13
ln |x − 1|”
, since
ln | 3√
x − 1| = ln |x − 1|1/3
Id
dx
1
3ln |x − 1| =
1
3
1
(x − 1)
d
dx(x − 1) =
1
3(x − 1)
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 23: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/23.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Chain Rule
d
dx(ln |x |) =
1
xand
d
dx(ln |g(x)|) =
g ′(x)
g(x)
Example Differentiate ln | sin x |.I Using the chain rule, we have
d
dxln | sin x | =
1
sin x
d
dxsin x
I
=cos x
sin x
Example Differentiate ln | 3√
x − 1|.I We can simplify this to finding d
dx
“13
ln |x − 1|”
, since
ln | 3√
x − 1| = ln |x − 1|1/3
Id
dx
1
3ln |x − 1| =
1
3
1
(x − 1)
d
dx(x − 1) =
1
3(x − 1)
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 24: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/24.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Chain Rule
d
dx(ln |x |) =
1
xand
d
dx(ln |g(x)|) =
g ′(x)
g(x)
Example Differentiate ln | sin x |.I Using the chain rule, we have
d
dxln | sin x | =
1
sin x
d
dxsin x
I
=cos x
sin x
Example Differentiate ln | 3√
x − 1|.I We can simplify this to finding d
dx
“13
ln |x − 1|”
, since
ln | 3√
x − 1| = ln |x − 1|1/3
Id
dx
1
3ln |x − 1| =
1
3
1
(x − 1)
d
dx(x − 1) =
1
3(x − 1)
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 25: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/25.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Substitution
Z1
xdx = ln |x |+ C and
Zg ′(x)
g(x)dx = ln |g(x)|+ C
Example Find the integral Zx
3− x2dx .
I Using substitution, we let u = 3− x2.
du = −2x dx , x dx =du
−2,
I Zx
3− x2dx =
Z1
−2(u)du
I
=−1
2ln |u|+ C =
−1
2ln |3− x2|+ C
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 26: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/26.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Substitution
Z1
xdx = ln |x |+ C and
Zg ′(x)
g(x)dx = ln |g(x)|+ C
Example Find the integral Zx
3− x2dx .
I Using substitution, we let u = 3− x2.
du = −2x dx , x dx =du
−2,
I Zx
3− x2dx =
Z1
−2(u)du
I
=−1
2ln |u|+ C =
−1
2ln |3− x2|+ C
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 27: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/27.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Substitution
Z1
xdx = ln |x |+ C and
Zg ′(x)
g(x)dx = ln |g(x)|+ C
Example Find the integral Zx
3− x2dx .
I Using substitution, we let u = 3− x2.
du = −2x dx , x dx =du
−2,
I Zx
3− x2dx =
Z1
−2(u)du
I
=−1
2ln |u|+ C =
−1
2ln |3− x2|+ C
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 28: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/28.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Using Substitution
Z1
xdx = ln |x |+ C and
Zg ′(x)
g(x)dx = ln |g(x)|+ C
Example Find the integral Zx
3− x2dx .
I Using substitution, we let u = 3− x2.
du = −2x dx , x dx =du
−2,
I Zx
3− x2dx =
Z1
−2(u)du
I
=−1
2ln |u|+ C =
−1
2ln |3− x2|+ C
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 29: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/29.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
To differentiate y = f (x), it is often easier to use logarithmic differentiation :
1. Take the natural logarithm of both sides to get ln y = ln(f (x)).
2. Differentiate with respect to x to get 1y
dydx
= ddx
ln(f (x))
3. We get dydx
= y ddx
ln(f (x)) = f (x) ddx
ln(f (x)).
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 30: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/30.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
Example Find the derivative of y = 4
√x2+1x2−1 .
I We take the natural logarithm of both sides to get
ln y = ln 4
rx2 + 1
x2 − 1
I Using the rules of logarithms to expand the R.H.S. we get
ln y =1
4ln
x2 + 1
x2 − 1=
1
4
hln(x2+1)−ln(x2−1)
i=
1
4ln(x2+1)− 1
4ln(x2−1)
I Differentiating both sides with respect to x, we get
1
y
dy
dx=
1
4· 2x
(x2 + 1)− 1
4· 2x
(x2 − 1)=
x
2(x2 + 1)− x
2(x2 − 1)
I Multiplying both sides by y, we get
dy
dx= y
h x
2(x2 + 1)− x
2(x2 − 1)
iI Converting y to a function of x, we get
dy
dx= 4
rx2 + 1
x2 − 1
"x
2(x2 + 1)− x
2(x2 − 1)
#
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 31: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/31.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
Example Find the derivative of y = 4
√x2+1x2−1 .
I We take the natural logarithm of both sides to get
ln y = ln 4
rx2 + 1
x2 − 1
I Using the rules of logarithms to expand the R.H.S. we get
ln y =1
4ln
x2 + 1
x2 − 1=
1
4
hln(x2+1)−ln(x2−1)
i=
1
4ln(x2+1)− 1
4ln(x2−1)
I Differentiating both sides with respect to x, we get
1
y
dy
dx=
1
4· 2x
(x2 + 1)− 1
4· 2x
(x2 − 1)=
x
2(x2 + 1)− x
2(x2 − 1)
I Multiplying both sides by y, we get
dy
dx= y
h x
2(x2 + 1)− x
2(x2 − 1)
iI Converting y to a function of x, we get
dy
dx= 4
rx2 + 1
x2 − 1
"x
2(x2 + 1)− x
2(x2 − 1)
#
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 32: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/32.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
Example Find the derivative of y = 4
√x2+1x2−1 .
I We take the natural logarithm of both sides to get
ln y = ln 4
rx2 + 1
x2 − 1
I Using the rules of logarithms to expand the R.H.S. we get
ln y =1
4ln
x2 + 1
x2 − 1=
1
4
hln(x2+1)−ln(x2−1)
i=
1
4ln(x2+1)− 1
4ln(x2−1)
I Differentiating both sides with respect to x, we get
1
y
dy
dx=
1
4· 2x
(x2 + 1)− 1
4· 2x
(x2 − 1)=
x
2(x2 + 1)− x
2(x2 − 1)
I Multiplying both sides by y, we get
dy
dx= y
h x
2(x2 + 1)− x
2(x2 − 1)
iI Converting y to a function of x, we get
dy
dx= 4
rx2 + 1
x2 − 1
"x
2(x2 + 1)− x
2(x2 − 1)
#
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 33: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/33.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
Example Find the derivative of y = 4
√x2+1x2−1 .
I We take the natural logarithm of both sides to get
ln y = ln 4
rx2 + 1
x2 − 1
I Using the rules of logarithms to expand the R.H.S. we get
ln y =1
4ln
x2 + 1
x2 − 1=
1
4
hln(x2+1)−ln(x2−1)
i=
1
4ln(x2+1)− 1
4ln(x2−1)
I Differentiating both sides with respect to x, we get
1
y
dy
dx=
1
4· 2x
(x2 + 1)− 1
4· 2x
(x2 − 1)=
x
2(x2 + 1)− x
2(x2 − 1)
I Multiplying both sides by y, we get
dy
dx= y
h x
2(x2 + 1)− x
2(x2 − 1)
iI Converting y to a function of x, we get
dy
dx= 4
rx2 + 1
x2 − 1
"x
2(x2 + 1)− x
2(x2 − 1)
#
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 34: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/34.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
Example Find the derivative of y = 4
√x2+1x2−1 .
I We take the natural logarithm of both sides to get
ln y = ln 4
rx2 + 1
x2 − 1
I Using the rules of logarithms to expand the R.H.S. we get
ln y =1
4ln
x2 + 1
x2 − 1=
1
4
hln(x2+1)−ln(x2−1)
i=
1
4ln(x2+1)− 1
4ln(x2−1)
I Differentiating both sides with respect to x, we get
1
y
dy
dx=
1
4· 2x
(x2 + 1)− 1
4· 2x
(x2 − 1)=
x
2(x2 + 1)− x
2(x2 − 1)
I Multiplying both sides by y, we get
dy
dx= y
h x
2(x2 + 1)− x
2(x2 − 1)
i
I Converting y to a function of x, we get
dy
dx= 4
rx2 + 1
x2 − 1
"x
2(x2 + 1)− x
2(x2 − 1)
#
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 35: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/35.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Logarithmic differentiation
Example Find the derivative of y = 4
√x2+1x2−1 .
I We take the natural logarithm of both sides to get
ln y = ln 4
rx2 + 1
x2 − 1
I Using the rules of logarithms to expand the R.H.S. we get
ln y =1
4ln
x2 + 1
x2 − 1=
1
4
hln(x2+1)−ln(x2−1)
i=
1
4ln(x2+1)− 1
4ln(x2−1)
I Differentiating both sides with respect to x, we get
1
y
dy
dx=
1
4· 2x
(x2 + 1)− 1
4· 2x
(x2 − 1)=
x
2(x2 + 1)− x
2(x2 − 1)
I Multiplying both sides by y, we get
dy
dx= y
h x
2(x2 + 1)− x
2(x2 − 1)
iI Converting y to a function of x, we get
dy
dx= 4
rx2 + 1
x2 − 1
"x
2(x2 + 1)− x
2(x2 − 1)
#Annette Pilkington Natural Logarithm and Natural Exponential
![Page 36: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/36.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Summary of formulas
ln(x)
ln(ab) = ln a + ln b, ln(a
b) = ln a− ln b
ln ax = x ln a
limx→∞
ln x =∞, limx→0
ln x = −∞
d
dxln |x | =
1
x,
d
dxln |g(x)| =
g ′(x)
g(x)Z1
xdx = ln |x |+ CZ
g ′(x)
g(x)dx = ln |g(x)|+ C .
Annette Pilkington Natural Logarithm and Natural Exponential
![Page 37: Natural Logarithm and Natural Exponentialapilking/Math10560/Lectures/Lecture... · 2020-01-13 · Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x)](https://reader033.vdocument.in/reader033/viewer/2022041804/5e539d0c42b1df34aa59c5dc/html5/thumbnails/37.jpg)
Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation summaries
Summary of methods
Logarithmic Differentiation(Finding formulas for inverse functions)Finding slopes of inverse functions (using formula from lecture 1).Calculating LimitsCalculating DerivativesCalculating Integrals (including definite integrals)
Annette Pilkington Natural Logarithm and Natural Exponential