5.2 LOGARITHMIC FUNCTIONS
Below is a graph of y  2 x , which is a one-to-one function (it passes the horizontal line test). Therefore,
it has an inverse. Interchanging x and y gives x  2 y , the inverse of y  2 x . To graph x  2 y , plot
points and reflect the graph of y  2 x around the line y  x .
y = 2x
8
y=x
6
x = 2y
4
2
5
10
15
-2
Since x  2 y is the inverse of y  2 x , we should find an equation for x  2 y by solving for y. However,
we cannot solve the equation for y with the methods we have presented up to now. Therefore, we
need a new definition to be able to solve for y:
Definition:
Logarithm – for all positive number a , with a  1 , and all positive number x ,
x  ay
means the same as
y  log a x .
The abbreviation log is used for the word logarithm. Read log a x as “the logarithm of x to the base a”
or “the base a logarithm of x.” To remember the location of the base and the exponent in each form,
refer to the following:
Logarithmic Form
Exponential Form
y  log a x
x  ay
To keep it simple, we should remember the following: A logarithm is an exponent! The expression
log a x represents the exponent to which the base a must be raised to obtain x.
We can use logarithms to write exponential statements into logarithmic form, and logarithmic
statements into exponential form.
Exponential Form
Logarithmic Form
2  log3 9
93
2
2
1
   25
5
105  100,000
1
4 3 
64
2  log1/ 5 25
5  log10 100,000
1
3  log 4
64
Your Turn: Complete the chart:
Exponential Form
Logarithmic Form
2  32
1001/ 2  10
5
2
 log8 4
3
4  log 6
1
1296
A logarithmic equation is an equation with a logarithm in at least one term. We solve logarithmic
equations of the form log a b  k for any of the three variables by first writing the equation in
exponential form.
Example 1 – Solve each equation:
a)
2  log 4 x
b) log x 3  2
c) log1/ 2  3x  1  2
d) log 49 3 7  x
Your turn:
a)
c) log1/ 2  2 x  5  2
3  log3 x
b) log x 5  4
d) log36 4 6  x
Two Properties of Logarithms:
1. For any positive real number b, with b  1, log b b  1
2. For any positive real number b, with b  1, log b 1  0
Example 2 – Use the properties of logarithms to evaluate each logarithm:
a)
log 7 7
b) log 9 1
c) log
2
2
d) log 0.2 1
Graphing Logarithmic Functions
By writing a logarithmic function in exponential form, we can then plot points and graph the logarithmic
function. It is easier to choose values for y and find the corresponding values of x. Plotting the points in
a table of ordered pairs and connecting them with a smooth curve gives the graph of a typical
logarithmic function. Be careful to write the x- and y-values in the correct order!
Example 3 – Graph each function:
a)
f ( x)  log 2 x
b)
g ( x)  log1/ 2 x
Things to note about the graph of G( x)  log a x
1.
The graph contains the point 1, 0 
2. When a  1 , the graph will rise from left to right, from the 4th Quadrant to the 1st Quadrant.
3. When 0  a  1 , the graph will fall from left to right, from the 1st Quadrant to the 4th
Quadrant.
4. The graph will approach the y-axis, but never touch it. (The y-axis is an asymptote.)
5. The domain is  0,   , and the range is  ,   .
Example 4 – An Application
The function defined by
f ( x)  27  1.105log10  x  1
Approximates the barometric pressure in inches of mercury at a distance of x miles from the eye of a
typical hurricane. (Source: Miller, A. and R. Anthes, Meterology, Fifth Edition, Charles E. Merrill
Publishing Comopany, 1985.)
a) Approximate the pressure 9 miles from the eye of the hurricane.
b) Approximate the pressure 99 miles from the hurricane.
Example 5 – An Application
A population of mites in a laboratory is growing according to the function defined by
P(t )  80log10  t  10
where t is the number of days after a study is begun.
a) Find the number of mites at the beginning of the study.
b) Find the number of mites present after 90 days.
c) Find the number of mites present after 990 days.
Example 6 – Converting from Logarithmic Form to Exponential Form
Write each equation in its equivalent exponential form:
a)
2  log 5 x
b)
3  logb 64
c)
log3 7  y
Example 7 – Converting from Exponential Form to Logarithmic Form
Write each equation in its equivalent logarithmic form:
a) 12 2  x
b)
b3  8
c)
ey  9
Example 8 – Evaluating Logarithms
A)
Evaluate each logarithmic function:
a)
f  x   log2 x
f  32  ?
c)
f  x   log4 x
f  2  ?
b)
f  x   log3 x
f 1  ?
d)
f  x   log10 x
 1 
f
?
 100 
B)
Evaluate:
a)
log 2 16
b)
log3 9
c)
log 25 5
THE COMMON LOGARITHM
The logarithmic function with base 10 is called the common logarithm. The function f ( x)  log10 x is
usually expressed as f ( x)  log x .
THE NATURAL LOGARITHM
The logarithmic function with base e is called the natural logarithm. The function f ( x)  log e x is
usually expressed as f ( x)  ln x .
Example 9 – Evaluating Common and Natural Logarithms
Evaluate:
a) log1000
b)
ln e7
d)
log107.1
ln e 1
1
e3
f)
log
c)
e) ln
1
10
INVERSE PROPERTIES OF LOGARITHMS
b logb x  x
(In words: b, raised to the logarithm with base b of a number, equals that number.)
Example 10 – Evaluating an exponential using inverse property of logarithms
Evaluate:
a)
6log 6 9
b)
eln 5
10log 4
c)
Example 11 – Evaluating the Natural and Common Logarithmic Functions Using a Calculator
Evaluate:
a)
ln 2
b) log 0.3
c) ln  1
d)

log 1  2

ONE-TO-ONE PROPERTIES OF LOGARITHMS
If log b M  log b N , then M  N
Example 12 – Using the One-to-One Property
Solve for x:
a) log3 x  log3 12
b) log  2 x  1  log3x
c)
log 5  x 2  6   log 5 10