8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
8­3 Logarithmic Functions as Inverses
3­8
sesrevnI sa snoitcnuF cimhtiragoL
Objectives: Convert between exponential form and logarithmic form.
Find the inverse of an exponential function.
Apply logarithms to real world situations.
1
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Warm­up:
Find the value of x in each example.
1) 3x = 27
2) 5x = 625
3) 4x = 1024
4) 2x = 2048
We need an easier way to solve these problems, but how do you undo exponents?
2
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
How do you solve this equation?
3x + 7 = 52
3
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Inverse functions are two functions whose operations undo each other.
function
f(x) = 3x ­ 2
­1
f inverse
x + 2
(x) = 3 Find f(f ­1(4))
Find f ­1(f(­4))
4
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
5
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Find the inverse of an exponential function.
bx = y
switch the variables and solve for y
by = x
We run into the same problem, how do we solve for y?
We need an inverse function for exponents.
This function is called a logarithm.
6
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Logarithms are defined in the following way:
If bx = y, then logb y = x.
pronounced "log base b of y equals x"
Using the definition rewrite the following exponential functions in logarithmic form.
25 = 32
32 = 9
43 = 64
7
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Using the definition, rewrite these logarithmic functions in exponential form.
log5 625 = 4
log10 10000 = 4
loge 54.598 ≈ 4
8
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Evaluating Logarithmic Functions
9
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Evaluate each logarithm.
a. log64 (1/32) b. log9 27
c. log10 100
10
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
A COMMON LOGARITHM is a logarithm that uses base 10. You can write the common logarithm
log10 y as log y.
The log button on your calculator is used for base 10 calculations.
11
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Use your calculator to evaluate each logarithm to four decimal places.
a. log 9
b. log (3/7)
c. log (­10)
WHY does the third problem fail? Rewrite it in exponential form to make it easier to see.
12
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Finding Inverse Functions
How do we find the inverse of y = 3x?
Step 1: Write the exponential equation as a logarithm.
y = 3x is equal to log3 y = x
Step 2: Interchange x and y. (because we switch the domain and range to get the inverse)
log3 y = x
log3 x = y
Therefore, the inverse of y = 3x is y = log3 x.
13
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Try one!
Find the inverse of y = (1/2)x.
14
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Logarithms and Exponential Functions are inverses. Therefore, we know that the graph of a logarithm is the graph of an exponential function reflected over the line y = x.
15
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
16
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Step 2: Draw y = x.
17
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Step 2: Draw y = x.
Step 3: Choose a few points on y = 2x.
Reverse the coordinates and plot the points of y = log2 x. 18
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
We have graphed exponential functions using a t­table such as the ones below.
y = bx
x y
­1 1/b
0 1
1 b
It's inverse:
Inverse y = logb x
x y
1/b ­1
1 0
b 1
The points on the graph of y = logb x are:
(1/b, ­1), (1, 0) and (b, 1).
19
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Logarithmic Properties
f (x) = logb x
1) The Domain : { x | x > 0}
Range : { y | y = (All Reals)}
0 < b < 1
(b,1)
(1,0)
2) There are no y intercepts and the x intercept is 1.
(1/b,­1)
3) The y axis (x=0) is a vertical asymptote as x 0.
4) f(x) = logb x , 0 < b < 1, is an decreasing function and is one­to­one.
5) The graph of f(x) contains the points (b,1), (1,0), (1/b,­1).
20
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
Logarithmic Properties
f (x) = logb x
b > 1
(b,1)
1) The Domain : { x | x > 0}
Range : { y | y = (All Reals)}
(1,0)
(1/b,­1)
2) There are no y intercepts and the x intercept is 1.
3) The y axis (x=0) is a vertical asymptote as x 0.
4) f(x) = logb x , b > 1, is an increasing function and is one­to­one.
5) The graph of f(x) contains the points (b,1), (1,0), (1/b,­1).
21
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
22
8­3 Logarithmic Functions as Inverses 2011
April 22, 2011
homework
8.1 ­ 8.3 Review
Worksheet
8.1 - 8.3 Quiz
Monday!!!
Time to complete the teacher eval!!! ASK ME!!
23