Objective: Students will be able to write equivalent forms
for exponential and logarithmic functions, and can write,
evaluate, and graph logarithmic functions.
Vocabulary
 Logarithm – the exponent to which a specified base is
raised to obtain a given value. It is the inverse of an
exponent.
 Common logarithm – a logarithm with base 10
 Logarithmic function – the inverse of an exponential
function
Finding Logarithms
 Solve 2x=8 using mental math.
 Now solve 2x=512
 This problem would be much easier to solve if
you could do so by taking the “mental math”
out.

This inverse operation is called finding a logarithm.
From exponential to logarithmic Form
 You can write an exponential equation as a logarithmic
equation and vice versa.
Example 1
Write each exponential equation in logarithmic
form.
Exponential
Equation
35 = 243
1
2
25 = 5
104 = 10,000
6–1 =
ab = c
1
6
Logarithmic Form
Example 2
Write each logarithmic form in exponential
equation.
Logarithmic
Form
log99 = 1
log2512 = 9
log82 =
log4
1
16
1
3
= –2
logb1 = 0
Exponential
Equation
 A logarithm is an exponent, so the rules for
exponents also apply to logarithms.
Common Logarithms
 A logarithm with base 10 is called a
common logarithm. If no base is written
for a logarithm, the base is assumed to be
10. For example, log 5 = log105.
 Example 3: Evaluate each log in your
calculator. log 0.01
Example 4
 log 1000
Logarithmic Functions
 Because logarithms are the inverses of exponents, the
inverse of an exponential function, such as y = 2x, is a
logarithmic function, such as y = log2x.
You may notice that the domain
and range of each function are
switched.
The domain of y = 2x is all real
numbers (R), and the range is
{y|y > 0}. The domain of
y = log2x is {x|x > 0}, and the
range is all real numbers (R).
Graphing Logarithmic Functions
Example 5: Use the x-values {–2, –1, 0, 1, 2}.
Graph the function and its inverse. Describe the
domain and range of the inverse function.
f(x) = 1.25x
Graph f(x) = 1.25x by using a
table of values.
x
f(x) = 1.25x
–2
–1
0
1
2
Example 5 continued…
–1
 Graph the inverse, f (x) = log1.25x, by
using a table of values.
x
f–1(x) = log1.25x
 Domain:
 Range:
Example 6
Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its
inverse. Describe the domain and range of the inverse
function.
 f(x) = (
–2
x
f(x) =(
)
)
x
–1
0
1
2
x
x
f –1(x) =log
1
2
x
4
2
1
–2
–1
0
1
2
Your turn…
Example 7: Use x = –2, –1, 1, 2, and 3 to graph f(x) = (.75)x
Then graph its inverse. Describe the domain and range of the
inverse function.
–2
x
–1
1
2
1
2
3
x
f(x) =
x
f–1(x) = log
x
–2
–1
3
Calculating Logarithms other than base 10
 Example 8: log7 343 =
 Example 9: log3 (
)=
Your turn…
 Example 10: log.5 .25 =
Homework for tonight
 Homework # ____
 Textbook pg. 253 # 18 – 26 even, 29, 30