1. 9.1, 9.3 Exponents and Logarithms

2. Graphs of Exponential Functions
The graph of f(x) = bx has a characteristic shape.
If b > 1, the graph goes uphill
If 0 < b < 1, the graph goes downhill
Domain is (–∞, ∞). Range is (0, ∞)
Unless translated the graph has a y-intercept of (0,1) 24

3. Definition of a Logarithm
A logarithm, or log, is defined in terms of an exponent:
If 52=25 then log525=2
You can say that the log is the exponent we put on 5 to get 25
If bx=a, then logba=x

4. Logarithmic Functions
x = 2y is an exponential equation.
If we solved for “y” we would get a logarithmic equation.
Here are the parts of each type of equation:
Exponential Equation x = 2y
Logarithmic Equation
y = log2 x
exponent
/logarithm
base
number

5. Example: Solve loga64 = 2 a2 = 64 Example : Solve log5 x = 3
Rewrite in exponential form!
loga64 = 2
base
number
exponent
a2 = 64
a = + 8 → a = 8
Example : Solve log5 x = 3
Rewrite in exponential form:
53 = x
x = 125

6. How do you graph a logarithmic function?
Re-write it as an exponential function and make a T-chart:
Example: Graph y = log3 x
Rewrite as: x = 3y
y = 3x
x y
1/9
1/3 1 3 9 -2 -1 1 2
y = log3 x

7. Graphs of Logarithmic Functions
The graph of f(x)=logbx has a characteristic shape.
The domain of the function is {x | x >0}
Unless translated, the graph has an x-intercept of 1.
Note the domain and range! -1 1 2 3 4 5 6

8. The logarithm with base 10 is called the common logarithm (this is the one your calculator evaluates with the LOG button)
The logarithm with base e is called the natural logarithm (this is the one your calculator evaluates with the LN button)

9. Examples. Evaluate each: a. log8 84 b. 6[log6 (3y – 1)]
logb bx = x
log8 84 = 4
blogb x = x
6[log6 (3y – 1)] = 3y – 1
Here are some IMPORTANT logarithm properties:
1. loga 1 = because a0 = 1
2. loga a = because a1 = a
3. loga ax = x because ax = ax