Solving Exponential Equations…
How do we solve these equations for x?
a. 10 x  100
x
10
 40
b.
x
10
 75
c.
Section 4.4 Logarithmic Functions
1.
2.
3.
4.
5.
6.
7.
Definition
Special Logarithms : Base 10 and Base e
Change of Base formula
Inverse Function
Graphing the log function
More log properties
Solving Equations
I. Definition of Logarithmic Functions
A logarithmic function with base a is denoted:
y  log a x
where a > 0 and a ≠ 1 and is defined by
y  log a x

The exponential form of
if and only if
y  log a x is a y  x
The logarithmic form of a y  x
is
y  log a x
What power of a gives you the value x ?
The answer:
x  ay
log a x
Practice : Rewrite the expression
Write the exponential expression into an equivalent logarithmic form.
16  4
125  5
x
x
Ask: What power of 4 produces the
number 16 ?
Answer: The logarithm of 16, base 4
Ask: What power of ______ produces the
number ____?
Answer: The logarithm of ____, base _____.
Practice : Rewrite the expression
Write the logarithmic expression into an equivalent exponential form.
log 2 16  x
Find the exact value (without a calculator).
log 3 9
3 raised to what power ___ equals 9 ?
log 2 1
2 raised to what power ___ equals 1 ?
log 1/ 2 16
½ raised to what power ___ equals 16 ?
2. Special Logarithms - Common Log
y  log x if and only if x  10 y
If a base is not indicated, it is understood to be 10.
Example:
y  log( 40)
is equivalent to
10  40
y
2. Special Logarithms – Natural Log
natural logarithm function is expressed
using the special symbol ln (logarithmus naturalis),
instead of the log symbol
y  ln x if and only if x  e y
Example:
y  ln 5
is equivalent to
e 5
y
Practice:
Solve exponential equations
and
evaluate log expressions
Review: #58
log 1/10 10 
log 12 12 
p. 284, #59, 60, 62, 63,66 Solve exponential
expression
p. 297 #33-37,39,41-43 Determine the value of the
logarithmic expression.
3. Change-of-Base Formula
log M ln M
log a M 

log a
ln a
Your calculator can compute only base 10 and base e.
Use this formula to get an approximation for a logarithm
to base neither 10 nor e.
Example.
Find an approximation for
log 2 (5)
4. Inverse Properties of Logarithmic
and Exponential Functions
The Logarithmic and Exponential Functions
are inverses of each other.
Inverse Property of g ( x)  b x and f ( x)  log b x
g ( f ( x))  b logb x  x and f ( g ( x))  log b b x  x
Example of the relationship: Let g ( x)  2
x
and f ( x)  log 2 x
5. Graphing Logarithmic Functions
6
Exponential functions and log
functions are inverse functions
of each other.
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
1 2 3 4 5 6
-2
-3
-4
f (x)  a
Domain:
Range:
Key
Points:
Asymptotes:
x
1
-5
f (x) -6log a x

5. A) Domain of a logarithmic function
Determine the domain for these functions.
f ( x)  log 4 x  2  1

f ( x)  log 3 2 x  8
2
 1 
f ( x)  ln 

 x 1 

Practice:
Worksheet 4.3-4.4
and
p. 297 #46-56 (even),
6) A Special Property of Logarithms
If log a M  log a N then M = N
and if M = N then log a M  log a N
NOTE:
We typically use base e
(natural log)
when applying the log to
both sides of the
equation.
7. Solving Logarithmic Equations.
Logarithmic Equations(always check answer against domain of problem)
1. Equal base on each side:
log 4 (2 x  8)  log 4 (6  2 x)
Use the property:
log a M  log a N then M = N
2. Constant on one side, logarithm on other:
log 4 (2x  8)  15
Use the definition:
y  log a x
a x
y
CHECK IT! Logarithms are only defined for positive real numbers!
Exclude solutions that produce logarithm of a number
8. Solving Exponential Equations.
Exponential Equations.
1. Equal base on each side:
Use the property:
3 x 4
2
2
x2
If au  av , then u  v
2. Constant on one side, exponential on other. Two ways to solve.
a) Method 1 :
e 4
3x
b) Method 2:
3x
2
1
4
3. Quadratic in form:
Use the definition:

ay  x
y  log a x
Take ln (natural log) of each side.
We will look at this method after Section 4.5
e  4e  3  0
2x
x
9. Application
The formula
A  15.9e
kx
models the population of Florida, A, in millions, x years after 2000.
Suppose the population is 16.3 million in 2001.
a) Determine the population of Florida in the year 2010.
b) When will the population reach 25.2 million ?