1. 5.2 Logarithmic Functions & Their Graphs
Goals—
Recognize and evaluate logarithmic functions with base a
Graph Logarithmic functions
Recognize, evaluate, and graph natural logs
Use logarithmic functions to model and solve real-life problems.

2. Is this function one to one?
Must pass the horizontal line test.
f(x) = 3x
Is this function one to one?
Yes
Does it have an inverse?
Yes

3. Logarithmic Function of base “a”
Definition:
Logarithmic function of base “a” -
For x > 0, a > 0, and a  1,
y = logax if and only if x = ay
Read as “log base a of x”
f(x) = logax is called the logarithmic function of base a.

4. The most important thing to remember about logarithms is…

5. a logarithm is an exponent.

6. Therefore, all logarithms can be written as exponential equations and all exponential equations can be written as logarithmic equations.

7. Write the logarithmic equation in exponential form
34 = 81
163/4 = 8
Write the exponential equation in logarithmic form
82 = 64
4-3 = 1/64
log 8 64 = 2
log4 (1/64) = -3

8. Evaluating Logs 2y = 32 2y = 25 y = 5 Think: y = log232 f(x) = log232
Step 1- rewrite it as an
exponential equation.
f(x) = log42
4y = 2
22y = 21
y = 1/2
2y = 32
f(x) = log10(1/100)
Step 2- make the bases the same.
10y = 1/100
10y = 10-2
y = -2
2y = 25
f(x) = log31
Therefore,
y = 5
3y = 1
y = 0

9. Evaluating Logs on a Calculator
You can only use a calculator when the
base is 10
Find the log key on your calculator.

10. Why? log 10 = 1 log 1/3 = -.4771 log 2.5 = .3979 log -2 = ERROR!!!
Evaluate the following using that log key.
log 10 = 1
log 1/3 =
log 2.5 = .3979
log -2 = ERROR!!!
Why?

11. Properties of Logarithms
loga1 = 0 because a0 = 1
logaa = 1 because a1 = a
logaax = x and alogax = x
If logax = logay, then x = y

12. Simplify using the properties of logs
Rewrite as an exponent
4y = 1
Therefore, y = 0
log41=
log77 = 1
Rewrite as an exponent
7y = 7
Therefore, y = 1
6log620 = 20

13. Use the properties of logs to solve these equations.
log3x = log312
x = 12
log3(2x + 1) = log3x
2x + 1 = x
x = -1
log4(x2 - 6) = log4 10
x2 - 6 = 10
x2 = 16
x = 4

14. Review:
How do you find the inverse of a function?
Application of what you know…
What is the inverse of f(x) = 3x?
y = 3x
x = 3y
y = log3x
f-1(x) = log3x
Rewrite the exponential as a logarithm…

15. Find the inverse of the following exponential functions…
f(x) = 2x f-1(x) = log2x
f(x) = 2x f-1(x) = log2x - 1
f(x) = 3x f-1(x) = log3(x + 1)

16. Find the inverse of the following logarithmic functions…
f(x) = log4x f-1(x) = 4x
f(x) = log2(x - 3) f-1(x) = 2x + 3
f(x) = log3x – f-1(x) = 3x+6

17. Graphs of Logarithmic Functions
Graph g(x) = log3x
It is the inverse of y = 3x
Therefore, the table of values for g(x) will be the reverse of the table of values for y = 3x.
y = 3x x y -1
1/3 1 3 2 9
y= log3x x y
1/3 -1 1 3 9 2 
Domain? (0,)  
Range? (-,) 
Asymptotes? x = 0

18. Graphs of Logarithmic Functions
g(x) = log4(x – 3)
What is the inverse exponential function?
y= 4x + 3
Show your tables of values.   
y= 4x + 3 x y -1
3.25 4 1 7 2 19
y= log4(x – 3) x y
3.25 -1 4 7 1 19 2
Domain? (3,)
Range? (-,)
Asymptotes? x = 3

19. Graphs of Logarithmic Functions
g(x) = log5(x – 1) + 4
What is the inverse exponential function?
y= 5x-4 + 1 
Show your tables of values.  
y= 5x-4 + 1 x y 3
1.2 4 2 5 6 26
y= log5(x – 1) + 4 x y
1.2 3 2 4 6 5 26
Domain? (1,)
Range? (-,)
Asymptotes? x = 1

20. Natural Logarithmic Functions
The function defined by
f(x) = logex = ln x, x > 0
is called the natural logarithmic function.

21. Evaluating Natural Logs on a Calculator
Find the ln key on your calculator.

22. Why? ln 2 = .6931 ln 7/8 = -.1335 ln 10.3 = 2.3321 ln -1 = ERROR!!!
Evaluate the following using that ln key.
ln =
ln 7/ =
ln =
ln = ERROR!!!
Why?

23. Properties of Natural Logarithms
ln1 = 0 because e0 = 1
Ln e = 1 because e1 = e
ln ex = x and eln x = x
If ln x = ln y, then x = y

24. Use properties of Natural Logs to simplify each expression
Rewrite as an exponent
ey = 1/e
ey = e-1
Therefore, y = -1
ln 1/e= -1
2 ln e = 2
Rewrite as an exponent
ln e = y/2
e y/2 = e1
Therefore, y/2 = 1 and y = 2. 5
eln 5=

25. Graphs of Natural Log Functions
g(x) = ln(x + 2)
Show your table of values.
y= ln(x + 2) x y -2
error -1
.693 1
1.099 2
1.386  
Domain? (-2,)  
Range? (-,)
Asymptotes? x = -2

26. Graphs of Natural Log Functions
g(x) = ln(2 - x)
Show your table of values.
y= ln(2 - x) x y 2
error 1
.693 -1
1.099 -2
1.386  
Domain? (-2,)  
Range? (-,)
Asymptotes? x = -2