[email protected] mth55_lec-60_fa08_sec_9-3a_intro-to-logs.ppt 1 bruce mayer, pe chabot...

[email protected] • MTH55_Lec-60_Fa08_sec_9-3a_Intro-to-Logs.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.3a§9.3aLogarithmsLogarithms



Review §Review §

Any QUESTIONS About• §9.2 → Inverse Functions

Any QUESTIONS About HomeWork• §9.2 → HW-43

9.2 MTH 55



Logarithm → What is it?Logarithm → What is it?

Concept: If b > 0 and b ≠ 1, then

y = logbx is equivalent to x = by

Symbolically

x = by y = logbx

The exponent is the logarithm.

The base is the base of the logarithm.



Logarithm IllustratedLogarithm Illustrated

Consider the exponential function f(x) = 3x. Like all exponential functions, f is one-to-one. Can a formula for f−1 be found? Use the 4-Step Method

f −1(x) ≡ the exponent to which we must raise 3 to get x.

y = 3x x = 3y

y ≡ the exponent to which we must raise 3 to get x.

4-Step



Logarithm IllustratedLogarithm Illustrated

Now define a new symbol to replace the words “the exponent to which we must raise 3 to get x”:

log3x, read “the logarithm, base 3, of x,” or “log, base 3, of x,” means “the exponent to which we raise 3 to get x.”

Thus if f(x) = 3x, then f−1(x) = log3x. Note that f−1(9) = log39 = 2, as 2 is the exponent to which we raise 3 to get 9



Example Example Evaluate Logarithms Evaluate Logarithms

Evaluate:a) log381 b) log31 c) log3(1/9)

Solution:a) Think of log381 as the exponent to which we

raise 3 to get 81. The exponent is 4. Thus, since 34 = 81, log381 = 4.

b) ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. So, log31 = 0

c) To what exponent do we raise 3 in order to get 1/9? Since 3−2 = 1/9 we have log3(1/9) = −2



The Meaning of logThe Meaning of logaaxx

For x > 0 and a a positive constant other than 1, logax is the exponent to which a must be raised in order to get x. Thus,

logax = m means am = x

or equivalently, logax is that unique exponent for which



Example Example Exponential to Log Exponential to Log

Write each exponential equation in logarithmic form.

a. 43 64 b. 1

2

4

1

16c. a 2 7

Soln a. 43 64 log4 64 3

b. 1

2

4

1

16 log1 2

1

164

c. a 2 7 loga 7 2



Example Example Log to Exponential Log to Exponential

Write each logarithmic equation in exponential form

a. log3 243 5 b. log2 5 x c. loga N x

Soln a. log3 243 5 243 35

b. log2 5 x 5 2x

c. loga N x N ax




Find the value of each of the following logarithmsa. log5 25 b. log2 16 c. log1 3 9

d. log7 7 e. log6 1 f. log4

1

2 Solution

a. log5 25 y 25 5y or 52 5y y 2

b. log2 16 y 16 2y or 24 2y y 4




Solution (cont.)

d. log7 7 y 7 7y or 71 7y y 1

e. log6 1 y 1 6y or 60 6y y 0

f. log4

1

2y

1

24 y or 2 1 22 y y

1

2

c. log1 3 9 y 9 1

3

y

or 32 3 y y 2



Example Example Use Log Definition Use Log Definition

Solve each equation for x, y or z

a. log5 x 3 b. log3

1

27y

c. logz 1000 3 d. log2 x2 6x 10 1

a. log5 x 3

x 5 3

x 1

53 1

125

The solution set is 1

125

.

Solution



Example Example Use Log Definition Use Log Definition

Solution (cont.)

b. log3

1

27y

1

273y

3 3 3y

3 y

c. logz 1000 3

1000 z3

103 z3

10 z

The solution set is 3 .

The solution set is 10 .



Inverse Property of LogarithmsInverse Property of Logarithms

Recall Def: For x > 0, a > 0, and a ≠ 1,

In other words, The logarithmic function is the inverse function of the exponential function; e.g.

xaxa xxa

a loglog



ShowShowLogLogaaaax

x = = xx



Example Example Inverse Property Inverse Property

Evaluate:

SolutionRemember that log523 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, obtain



Basic Properties of LogarithmsBasic Properties of Logarithms

For any base a > 0, with a ≠ 1, Discern from the Log Definition

1. Logaa = 1

• As 1 is the exponent to which a must be raised to obtain a (a1 = a)

2. Loga1 = 0

• As 0 is the exponent to which a must be raised to obtain 1 (a0 = 1)



Graph Logarithmic FunctionGraph Logarithmic Function

Sketch the graph of y = log3x

Soln:MakeT-Table→

x y = log3x (x, y)

3–3 = 1/27 –3 (1/27, –3)

3–2 = 1/9 –2 (1/9, –2)

3–3 = 1/3 –1 (1/3, –1)

30 = 1 0 (1, 0)

31 = 3 1 (3, 1)

32 = 9 2 (9, 2)



Graph Logarithmic FunctionGraph Logarithmic Function

Plot the ordered pairs and connect the dots with a smooth curve to obtain the graph of y = log3x



Example Example Graph by Inverse Graph by Inverse Graph y = f(x) = 3x

Solution:Use Inverse Relationfor Logs & Exponentials

Reflect the graph of y = 3x in the line y = x to obtain the graph ofy = f−1(x) = log3x



Domain of Logarithmic FcnsDomain of Logarithmic Fcns

Recall that the• Domain of f(x) = ax is (−∞, ∞)

• Range of f(x) = ax is (0, ∞)

Since the Logarithmic function is the inverse of the Exponential function,• Domain of f−1(x) = logax is (0, ∞)

• Range of f−1(x) = logax is (−∞, ∞)

Thus, the logarithms of 0 and negative numbers are NOT defined.



Example Example Find Domain Find Domain

Find the domain of each function.

a. f x log3 2 x b. f x log3

x 2

x 1

Solution a.The Domain of a logarithmic function must be positive, that is,

2 x 0

2 x

Thus The domain of f is (−∞, 2).




Find the domain of each function.

b. f x log3

x 2

x 1

Solution b.The Domain of a logarithmic function must be positive, that is,

Need to Avoid Negative-Logs AND Division by Zero

x 2

x 1 0




Soln b. (cont.) b. f x log3

x 2

x 1

Set numerator = 0 & denominator = 0

Construct a SIGN CHART

x − 2 = 0 x + 1 = 0x = 2 x = −1

The domain of f is (−∞, −1)U(2, ∞).



Properties of Exponential and Properties of Exponential and Logarithmic FunctionsLogarithmic Functions

Exponential Function f (x) = ax

Logarithmic Function f (x) = loga x

Domain (0, ∞) Range (–∞, ∞)

Domain (–∞, ∞) Range (0, ∞)

x-intercept is 1 No y-intercept

y-intercept is 1 No x-intercept

x-axis (y = 0) is the horizontal asymptote

y-axis (x = 0) is the vertical asymptote



Properties of Exponential and Properties of Exponential and Logarithmic FunctionsLogarithmic Functions

Exponential Function f (x) = ax

Logarithmic Function f (x) = loga x

Is one-to-one, that is, logau = logav if and only if u = v

Is one-to-one , that is, au = av if and only if u = v

Increasing if a > 1 Decreasing if 0 < a < 1

Increasing if a > 1 Decreasing if 0 < a < 1



Graphs of Logarithmic FcnsGraphs of Logarithmic Fcns

f (x) = loga x (0 < a < 1)f (x) = loga x (a > 1)



Graph Logs by TranslationGraph Logs by Translation

Start with the graph of f(x) = log3x and use Translation Transformations to sketch the graph of each function

a. f x log3 x 2 b. f x log3 x 1

Also State the DOMAIN and RANGE and the VERTICAL ASYMPTOTE for the graph of each function




Solutiona. f x log3 x 2

• Shift UP 2

• Domain (0, ∞)

• Range (−∞, ∞)

• Vertical asymptote x = 0




Solution

• Shift RIGHT 1

• Domain (1, ∞)

• Range (−∞, ∞)

• Vertical asymptote x = 1

b. f x log3 x 1



WhiteBoard WorkWhiteBoard Work

Problems From §9.3 Exercise Set• 8, 18, 26, 38, 48

Logs &ExponentialsAre InverseFunctions



All Done for TodayAll Done for Today

Inventorof

Logarithms

Born: 1550 in Merchiston Castle, Edinburgh, Scotland

Died: 4 April 1617 in Edinburgh, Scotland

John Napier



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

[email protected] mth55_lec-60_fa08_sec_9-3a_intro-to-logs.ppt 1 bruce mayer, pe chabot...

Documents