Digital Lesson
Logarithmic Functions
For x  0 and 0  a  1,
y = loga x if and only if x = a y.
The function given by f (x) = loga x is called the
logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form:
y = loga x is equivalent to x = a y
A logarithm is an exponent!
A logarithmic function is the inverse function of an exponential
function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
2
Examples: Write the equivalent exponential equation
and solve for y.
Logarithmic
Equation
Equivalent
Exponential
Equation
y = log216
1
y = log2( )
2
y = log416
16 = 2y
1
= 2y
2
16 = 4y
y = log51
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
1=5y
Solution
16 = 24  y = 4
1
= 2-1 y = –1
2
16 = 42  y = 2
1 = 50  y = 0
3
The base 10 logarithm function f (x) = log10 x is called the
common logarithm function.
The LOG key on a calculator is used to obtain common
logarithms.
Examples: Calculate the values using a calculator.
Function Value
Keystrokes
Display
log10 100
2
log10( )
5
log10 5
log10 –4
LOG 100 ENTER
2
LOG ( 2  5 ) ENTER
– 0.3979400
LOG 5 ENTER
LOG –4 ENTER
0.6989700
ERROR
no power of 10 gives a negative number
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. loga ax = x and alogax = x
inverse property
4. If loga x = loga y, then x = y. one-to-one property
Examples: Solve for x: log6 6 = x
log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log79
7log79 = 9 property 3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
5
Graph f (x) = log2 x
Since the logarithm function is the inverse of the exponential
function of the same base, its graph is the reflection of the
exponential function in the line y = x.
x
y
y
=
2
y=x
x
2x
–2
1
4
–1
1
2
0
1
1
2
2
3
horizontal
asymptote y = 0
4
8
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y = log2 x
x-intercept
x
(1, 0)
vertical asymptote
x=0
6
Example: Graph the common logarithm function f(x) = log10 x.
x
f(x) = log10 x
1
100
–2
1
10
–1
1
2
4
10
0
0.301
0.602
1
by calculator
y
f(x) = log10 x
x
5
(0, 1) x-intercept
x=0
vertical
asymptote
–5
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
7
The graphs of logarithmic functions are similar for different
values of a.
f(x) = loga x (a  1)
Graph of f (x) = loga x (a  1)
1. domain (0, )
2. range (,)
3. x-intercept (1, 0)
4. vertical asymptote
x  0 as x  0 f ( x)  
5. increasing
6. continuous
7. one-to-one
8. reflection of y = a x in y = x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y-axis
vertical
asymptote
y
y = ax
y=x
y = log2 x
domain
x
x-intercept
(1, 0)
range
8
y
The function defined by
f(x) = loge x = ln x
(x  0, e  2.718281)
is called the natural
logarithm function.
y = ln x
x
5
–5
y = ln x is equivalent to e y = x
Use a calculator to evaluate: ln 3, ln –2, ln 100
Function Value
Keystrokes
ln 3
LN 3 ENTER
ln –2
LN –2 ENTER
ln 100
LN 100 ENTER
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Display
1.0986122
ERROR
4.6051701
9
Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x and eln x = x
inverse property
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
 
1
ln  2   ln e 2  2
e 
inverse property
e ln 20 20
inverse property
3 ln e  3(1)  3
property 2
ln 1  0  0
property 1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
10
t
Example: The formula R  112 e 8223 (t in years) is used to estimate
10
the age of organic material. The ratio of carbon 14 to carbon 12
in a piece of charcoal found at an archaeological dig is R  115 .
10
How old is it?
t
1 8223
1
e
 15
original equation
12
10
10
t
1
8223
e

multiply both sides by 1012
1000
t
1
8223
ln e
 ln
take the natural log of both sides
1000
t
1
 ln
inverse property
8223
1000
1 

t  8223  ln
  8223  6.907   56796
 1000 
To the nearest thousand years the charcoal is 57,000 years old.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
11