logarithmic functions. logarithm = exponent very simply, a logarithm is an exponent of ten that will...
TRANSCRIPT
Logarithmic FunctionsLogarithmic FunctionsLogarithmic FunctionsLogarithmic Functions
Logarithm = Exponent
Very simply, a logarithm is an exponent of ten that will produce the desired number.
Y = Log 100 means what is the exponent of 10 which will produce 100.Y = log .001 means what is the exponent of 10 which will produce .001.Note: when no base is indicated, the base is 10.
Y=logb a can be read “ y is the exponent of base b to produce a.”
3= log2 8 is read “3 is the exponent of 2 to produce 8”
2 = log4 16
42 = 16
22
1 12 log 2
4 4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
y = log2( )2
1= 2
y2
1
Write the equivalent exponential equation and solve for y.
1 = 5 yy = log51
16 = 4y y = log416
16 = 2yy = log216
SolutionEquivalent Exponential Equation
Logarithmic Equation
16 = 24 y = 4
2
1= 2-1 y = –1
16 = 42 y = 2
1 = 50 y = 0
Definition of a Logarithm
• A logarithm, or log, is defined in terms of an exponent.
• If b x=a, then logb a =x
If 5 2=25 then log5 25=2
Log5 25=2 is read “log base 5 of 25 is 2.”
– You might say the log is the exponent we apply to 5 to make 25
Log of a Product
• The log of a product is the sum of the logs of the factors
logbxy = logbx + logby
Log2512 = log2(8·64)
= log28 + log264
= 3 + 5 = 8
Log of a Quotient
The log of a quotient is the difference of the logs of the factors.
log log logb b bx x yy
Ex. 5 5 5125
log log 125 log 2525
3 2
1
Log of a Power
The log of a power is the product of the exponent and the log of the base.
logbxn = nlogbx
Ex: log 32 = 2log3
Use the properties of logs to simplify the following:
log 2 .3456 log 3 .4234log 5 .4985 log 7 .6002a
a a
a
Given:
34
log 15 log 42
5 2log log
3 7
log 7 log 2 5
a a
a a
a a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
For x 0 and 0 a 1, y = loga x if and only if x = a y.The function given by f (x) = loga x is called the logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y
A logarithmic function is the inverse function of an exponential function.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
log10 –4 LOG –4 ENTER ERROR
REMEMBER: no power of 10 gives a negative number
The logarithm function f (x) = log10 x is called the common logarithm function.
log10 100
log10 5
Function Value
Keystrokes
Display
LOG 100 ENTER 2
LOG 5 ENTER 0.69897005
2log10( ) – .3979400LOG ( 2 / 5 ) ENTER
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
Properties of Logarithms
1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. If loga x = loga y, then x = y. one-to-one property
INVERSE PROPERTIES:
The logarithm with base a of a raised to a power equals that power:
a raised to the logarithm with base a of a number equals that number
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
log xa a x
loga xa x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
One way to Graph f (x) = log2 xSince the logarithm function is the inverse of
the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.
83422110
–1
–22xx
4
1
2
1
y = log2 x
y = xy = 2x
(1, 0)x
y
x-intercept
horizontal asymptote y = 0
vertical asymptote x = 0
Graphing Log Functions
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16
One way to Graph f (x) = log2 xSince the logarithm function is the inverse of
the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.
83422110
–1
–22xx
4
1
2
1
y = log2 x
y = xy = 2x
(1, 0)x
y
x-intercept
horizontal asymptote y = 0
vertical asymptote x = 0
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17
Graph of the common logarithm function f(x) = log10 x.
by calculator
1
10
10.6020.3010–1–2f(x) = log10
x
10421x 1
100
y
x5
–5
f(x) = log10 x
x = 0 vertical asymptote
(0, 1) x-intercept
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18
The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)
3. x-intercept (1, 0)
5. increasing6. continuous7. one-to-one 8. reflection of y = a
x in y = x
1. domain
),0( 2. range
),(
4. vertical asymptote )(0 as 0 xfxx
Graph of f (x) = loga x (a
1)
x
y y = x
y = log2
x
y = a x
domain
range
y-axisverticalasymptote
x-intercept(1, 0)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19
Using the graphing calculator to graph the log
functions
• For other than base 10, use the following formula:
10
10
loglog
loga
xx
a
To graph the function
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
Graph the following functions:
2
3
4
5
( ) log
( ) log
( ) log
( ) log
f x x
f x x
f x x
f x x
2 3log logy x y x
4 5log logy x y x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22
The function defined by f(x) = loge x = ln x
is called the natural logarithm function.
Use a calculator to evaluate: ln 3, ln –2, ln 100
ln 3ln –2ln 100
Function Value
Keystrokes DisplayLN 3 ENTER 1.0986122
ERRORLN –2 ENTERLN 100 ENTER 4.6051701
y = ln x
(x 0, e .718281)
y
x5
–5
y = ln x is equivalent to e y
= x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23
Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
2
1ln
e 2ln 2 e inverse property
20lne 20 inverse property
eln3 3)1(3 property 2
00 1ln property 1
.
.