1. Logarithmic Functions and Graphs
Section 4.3
Logarithmic Functions and Graphs

2. Flashback
Consider the graph of the exponential function y = f(x) = 3x.
Is f(x) one-to-one?
Does f(x) have an
inverse that is a
function?
Find the inverse.

3. Inverse of y = 3x
f (x) = 3x
y = 3x
x = 3y

4. y= the power to which 3 must be raised in order to obtain x.
x = 3y
Now, solve for y.
y= the power to which 3 must be raised in order to obtain x.

5. Symbolically, y = log 3 x x = 3y Solve for y.
y= the power to which 3 must be raised in order to obtain x.
Symbolically, y = log 3 x
“The logarithm, base 3, of x.”

6. Logarithm For all positive numbers a, where a 1,
Logax is an exponent to which the base a
must be raised to give x.

7. All a log is . . . is an exponent!
Logarithmic Form
Exponential Form
Argument
(always positive)
All a log is is an exponent!

8. Logarithmic Functions
Logarithmic functions are inverses of exponential functions.
Graph: x = 3y or y = log 3 x
1. Choose values for y.
2. Compute values for x.
3. Plot the points and connect them with a
smooth curve.
* Note that the curve does not touch or cross
the y-axis.

9. Logarithmic Functions continued
Graph: x = 3y
y = log 3 x
(1/27, 3) 3
1/27
(1/9, 2) 2
1/9
(1/3, 1) 1
1/3 2 1 y
(9, 2) 9
(3, 1) 3
(1, 0)
(x, y)
x = 3y

10. Side-by-Side Comparison
f (x) = 3x
f (x) = log 3 x

11. Comparing Exponential and Logarithmic Functions

12. Logarithmic Functions
Remember: Logarithmic functions are inverses of exponential functions.

13. Asymptotes
Recall that the horizontal asymptote of the exponential function y = ax is the x-axis.
The vertical asymptote of a logarithmic function is the y-axis.

14. Logarithms Convert each of the following to a logarithmic equation.
a) 25 = 5x b) ew = 30

15. Example Convert each of the following to an exponential equation.
a) log7 343 = 3
log7 343 = = 343
b) logb R = 12
The logarithm is the exponent.
The base remains the same.

16. Finding Certain Logarithms
Find each of the following.
a) log b) log
c) log d) log

17. Log button on your calculator
Common Logarithm
Logarithms, base 10, are called common logarithms.
Log button on your calculator
is the common log *

18. Example a) log 723,456 b) log 0.0000245 c) log (4)
Find each of the following common logarithms on a calculator.
Round to four decimal places.
a) log 723,456
b) log
c) log (4)
Does not exist
ERR: nonreal ans
log (4)
4.6108 
log
5.8594
log 723,456
Rounded
Readout
Function Value

19. * ln button on your calculator
Natural Logarithms
Logarithms, base e, are called natural logarithms.
The abbreviation “ln” is generally used for natural logarithms.
Thus, ln x means loge x.
* ln button on your calculator
is the natural log *

20. Example a) ln 723,456 b) ln 0.0000245 c) ln (4)
Find each of the following natural logarithms on a calculator.
Round to four decimal places.
a) ln 723,456
b) ln
c) ln (4)
Does not exist
ERR: nonreal ans
ln (4)   ln
ln 723,456
Rounded
Readout
Function Value

21. Changing Logarithmic Bases
The Change-of-Base Formula
For any logarithmic bases a and b,
and any positive number M,
Use change of base formula when you have a logarithm that is not base 10 or e.

22. Example Find log6 8 using common logarithms.
Solution: First, we let a = 10, b = 6, and M = 8. Then we substitute into the change-of-base formula:

23. Example We can also use base e for a conversion.
Find log6 8 using natural logarithms.
Solution: Substituting e for a, 6 for b and 8 for M, we have

24. Properties of Logarithms

25. Graphs of Logarithmic Functions
Graph: y = f(x) = log6 x.
Select y.
Compute x. 2
1/36 1
1/6 3
216 2 36 1 6 y
x, or 6 y

26. Example Graph each of the following.
Describe how each graph can be obtained from the graph of y = ln x.
Give the domain and the vertical asymptote of each function.
a) f(x) = ln (x  2)
b) f(x) = 2  ¼ ln x
c) f(x) = |ln (x + 1)|

27. Graph f(x) = ln (x  2) 1.099 5 0.693 4 3 0.693 2.5 1.386 2.25 f(x)
The graph is a shift 2 units right.
The domain is the set of all real numbers greater than 2.
The line x = 2 is the vertical asymptote.
1.099 5
0.693 4 3
0.693
2.5
1.386
2.25
f(x) x

28. Graph f(x) = 2  ¼ ln x 1.598 5 1.725 3 2 1 2.173 0.5 2.576 0.1 f(x) x
The graph is a vertical shrinking, followed by a reflection across the x-axis, and then a translation up 2 units.
The domain is the set of all positive real numbers.
The y-axis is the vertical asymptote.
1.598 5
1.725 3 2 1
2.173
0.5
2.576
0.1
f(x) x

29. Graph f(x) = |ln (x + 1)| 1.946 6 1.386 3 0.693 1 0.5 f(x) x
The graph is a translation 1 unit to the left.
Then the absolute value has the effect of reflecting negative outputs across the x-axis.
The domain is the set of all real numbers greater than 1.
The line x = 1 is the vertical asymptote.
1.946 6
1.386 3
0.693 1
0.5
f(x) x

30. Application: Walking Speed
In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function
w(P) = 0.37 ln P

31. Application: Walking Speed continued
The population of Philadelphia, Pennsylvania, is
1,517,600.
Find the average walking speed of people living in
Philadelphia.
Since 1,517,600 = thousand,
we substitute for P, since P is in thousands:
w(1517.6) = 0.37 ln
 2.8 ft/sec.
The average walking speed of people living in Philadelphia is about 2.8 ft/sec.