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Logarithmic Functions
We know:
23 = 8
and
But, for what value of x does 2x = 10?
24 = 16
Since 10 is between 8 and 16, x must be between 3 and 4.
To solve for an exponent, mathematicians defined logarithms.

Definition of Logarithm
Logarithmic Form
Exponential Form
if and only if
y=bx
x = log b y
Power
Base
b and y are positive real numbers
and b ≠ 1
So, 2x = 10, from our little example, can be written as:
x = log 2 10
Name the power: x
Name the base:
2

Example 1. Rewrite each equation in exponential form.
2
 3 =9
a. log3 9 = 2
First, write the Base.
y=bx
if and only if
x = log b y
Then write the power.
This equals to what’s left over.
b. log8 1 = 0
0
 8 =1
−2
1
 1 
c. log5     2  5 
25
 25 

Example 2. Evaluate the expression.
y=bx
a. log4 64 Which piece is missing?
if and only if
x = log b y
When evaluating logs, the solution is the
power that makes the log a true statement.
log4 64 = ? Rewrite the equation in exponential form.
?
 4 = 64  log4 64 = 3
1  ? 1
1

2   log 2   = −3
b. log2 0.125  log 2  
8
8
8
?
1

c. log1/4 256     256
 log1/4 256 = −4
4
1
?
d. log32 2  32 = 2  log32 2 
5

Example 2. Evaluate the expression using the calculator.
MATH A allows you to enter the base and the number to
get the answer.
a. log4 64
MATH A 4 (64) = 3
b. log2 0.125
MATH A 2 (0.125) = -3
c. log1/4 256
MATH A 1/4 (256) = -4
d. log32 2
MATH A 32 (2) =
1
5

Look at the definition of a logarithm again.
y=bx
if and only if
x = log b y
Exponential and Logarithmic functions are INVERSES of
each other!!!
This means that the domain and range switch places!!
Logarithms always have a RANGE of all real
numbers and a limited domain.
Logarithms have vertical asymptotes.
Exponential expressions always have a DOMAIN of
all real numbers and a limited range.
Exponentials have horizontal asymptotes.
𝟐<𝒙<∞
ALWAYS All Reals!!
𝒙=𝟑
None
𝒙=𝟑
𝟐<𝒙<∞
Never
−∞
∞
NOTE: Your calculator cannot draw the vertical asymptote, so it
appears as though the graph stops at x = 2; it does not!! The
graph continues down forever; the range is all real numbers. Keep
this in mind at all times!!
ALWAYS All Reals!!
𝟎<𝒚<∞
None
𝒚=𝟔
None
−∞ < 𝒙 < ∞
𝟎
−∞
Never
∞
Remember: Exponential functions are INVERSES of logarithms,
so the domains & ranges switch. The domain of an exponential
function is always all real numbers.
This makes
the domain
all real #’s
𝑦 = −5
This makes
the range all
real #’s
𝑥 = −2
𝑦 = 4999.98 ∗ 1.035𝑥
$8376.7