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Lecture 5.7
Contemporary Mathematics
Instruction: Logarithms, Logarithmic Functions, and Logarithmic Properties
A logarithm is the exponent to which some pre-chosen base must be raised in order to
obtain a specified number called the argument. In symbols, log b a = x means b x = a where b is
a positive number not equal to one.
If b x = a where b > 0 and b ≠ 1 , then log b a = x . The notation log b a is read
"logarithm base b of a" and equals the exponent to which b must be raised in
order to obtain a. By definition, then, a logarithm is an exponent.
Consider the exponential equation x = b y . In this equation, b raised to the y exponent yields x, a
power of b. The number b, therefore, is the base and y is the logarithm to base b of x. This
equation can be written alternatively:
y = log b x
This form of the equation depicts y as the logarithm to base b of x (the specified power of the
base called the argument of the logarithm) and can be expressed as a function f : x 6 log b x .
This logarithmic function is a function that maps a power of some positive base not equal to one
to the exponent that yields the power. Since all powers of a positive base are themselves
positive, the domain includes only positive numbers.
Let D and Y be the domain and co-domain respectively. Let Y = \ , and let
D = { x | x ∈ \, x > 0} . A logarithmic function with base b, where b > 0 and
b ≠ 1 , is a function that maps x to the exponent that yields x when applied to b
and is denoted f : x 6 log b x .
Special notation is used for logarithms to the base 10 and for logarithms to the base e.
Usually the logarithm's base is written as a subscript to the "log" symbol. With base ten
logarithms the base is not written but simply understood to be ten. Thus, " log100 " reads,
"logarithm to the base ten of one hundred." Base e logarithms are called natural logarithms.
With natural logarithms, the "log" symbol is replaced by "LN" or "ln" and the base is understood
to be e. Thus, "ln 2" reads, "logarithm to the base e of two" or "natural log of two."
Evaluating logarithmic functions sometimes requires a calculator. Most calculators are
programmed to evaluate base ten logarithms and natural logarithms. For our purposes, however,
a calculator is not necessary; although, students must be aware of common powers and roots of
the natural numbers. Consider log 2 8 . This expression reads, "logarithm to the base two of
eight." The logarithm equals 3 because 23 = 8 (eight is the third power of two).
Consider the logarithmic function M : x 6 log 9 x . Using this function we will map
1 9, 1, 3, 9, and 81.
Lecture 5.7
M :1 9 6 log 9 1 9 = −1 because 9−1 = 1 9.
M :1 6 log 9 1 = 0
because 90 = 1.
M : 3 6 log 9 3 = 1 2
because 91 2 = 9 = 3.
M : 9 6 log 9 9 = 1
because 91 = 9.
M : 81 6 log 9 81 = 2
because 92 = 81.
The mappings above display some general characteristics of logarithmic functions. First,
we note that as the x-values increased, the functional values increased, but the increase was
relatively slow. The x-values increased from 1 9 to 81, but the y-values only increased from –1
to 2 respectively. In general, logarithmic functions increase if b > 1 and decrease if b < 1 . We
recall that exponential functions also increase if b > 1 and decrease if b < 1 . However, the rate of
increase/decrease of logarithmic functions is typically must slower than exponential functions.
Second, we note that M mapped one to zero. Indeed, logarithmic functions always map
one to zero since log b 1 = 0.
Third, we note M can only map positive values. In fact, the definition states that the
domain includes only positive real numbers. Moreover, the graph of a logarithmic function
approaches the y-axis as x-values decrease toward zero.
Figure 1 shows the graph of a logarithmic function where b > 1 . Figure 2 shows the
graph of an logarithmic function where 0 < b < 1 . For both graphs, the y-axis serves as a vertical
asymptote, which the graph will approach closely but never actually touch.
y
y
(1, 0)
(1, 0)
x
Figure 1
x
Figure 2
Logarithms share some properties. Two important properties are detailed below.
Property 1: For all real numbers p, log b b p = p where b > 0, b ≠ 1, and a > 0.
Property 1 is a direct application of the definition of a logarithm. A logarithm is the exponent
applied to the base that yields the argument.
Property 2: For all real numbers p, log b a p = p ⋅ log b a where b > 0, b ≠ 1, and a > 0.
Property 2 is an application of the exponent property that states ( b r ) = b rp . Properties 1 and 2
p
are important for solving exponential equations. Consider the question, "What x-value is
Lecture 5.7
mapped to 81 by the exponential function f : x 6 27 x ?" In other words, what is the solution to
the equation below?
27 x = 81
To solve, we can make both sides of the equation the argument of a logarithm as below.
log 3 27 x = log 3 81
Applying property 2, we can write the left side of the equation as a pair of factors as below.
x ⋅ log 3 27 = log 3 81
Since 27 and 81 are both recognizable powers of three, we can rewrite them as such and apply
property 1 as below.
x ⋅ log 3 33 = log 3 34
x⋅3 = 4
Finally, we solve the resulting linear equation.
3x 4
=
3 3
x=4 3
Thus, the function f maps four-thirds to eighty-one.
Application Exercise 5.7
Problems
#1
Astronomers use the distance modulus of a star as a measure of distance from Earth. The function
M : d 6 5log10 ( d ) − 5 maps d, a star's distance from Earth in parsecs to M, the star's distance
modulus. If a star is 1,000 parsecs from Earth, what is its distance modulus?
#2
Use the function g : x 6 log 3 x to map 1 9, 1, 3, 9, and 729.
#3
Determine what x-value is mapped to 256 by the exponential function f : x 6 8x .
#4
Sketch the graph of the function L : x 6 log 5 x .
#1 M = 10
#2 1 9 6 −2 , 1 6 0 , 3 6 1 , 9 6 2 , 729 6 6
#3 x = 8 3
#4
domain: (0,∞)
vertical asymptote: x = 0
x-intercept: (1,0)
Assignment 5.7
Problems
#1
Write a logarithmic function that decreases as x-values increase.
#2
Use the function h : x 6 log8 x to map 1 64, 1, 2, 8, and 64.
#3
Determine what x-value is mapped to 8 by the exponential function f : x 6 4 x .
#4
Sketch the graph of the function g : x 6 log 2 x .
#5
Consider P : x 6 ln ( x ) . Determine if P is bijective.