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David Terr
Ph.D. Math, UC Berkeley

3.2. Logarithmic Functions

Closely related and of equal importance to exponential functions are another class of functions known as logarithmic functions, or more simply logarithms. A logarithmic function is simply the functional inverse of an exponential function. In other words, the base-b logarithm of a positive real number x, written as logbx, is equal to the unique number y such that by = x. This definition works because exponential functions are one-to-one, meaning that for b and x fixed, the exponent y such that by = x is unique.

The most commonly used logarithms used to be base-10 logarithms. (Nowadays, natural logarithms are more commonly used, but we will not discuss natural logarithms in this syllabus, since it is better to wait until one learns calculus.) For this reason, base-10 logarithms are known as common logarithms, and the subscript 10 is omitted from their notation, i.e. instead of writing log10x we simply write log x.

It is easy to compute common logarithms of powers of 10. The following table lists the common logarithms of the powers of 10 listed in the tables in Section P.2:

name x SN log x
one-trillionth
0.000000000001
10-12
-12
one hundred-billionth
0.00000000001
10-11
-11
one ten-billionth
0.0000000001
10-10
-10
one billionth
0.000000001
10-9
-9
one hundred-millionth
0.00000001
10-8
-8
one ten-millionth
0.0000001
10-7
-7
one millionth
0.000001
10-6
-6
one hundred-thousandth
0.00001
10-5
-5
one ten-thousandth
0.0001
10-4
-4
one thousandth
0.001
10-3
-3
one hundredth
0.01
10-2
-2
one tenth
0.1
10-1
-1
one
1
100
0
ten
10
101
1
one hundred
100
102
2
one thousand
1,000
103
3
ten thousand
10,000
104
4
one hundred thousand
100,000
105
5
one million
1,000,000
106
6
ten million
10,000,000
107
7
one hundred million
100,000,000
108
8
one billion
1,000,000,000
109
9
ten billion
10,000,000,000
1010
10
one hundred billion
100,000,000,000
1011
11
one trillion
1,000,000,000,000
1012
12

Nowadays, logarithms to arbitrary bases are easy to compute by means of scientific calculators, but in the old days (prior to around 1980), elaborate tables of logarithms known as log tables were used for this purpose. Below is a brief table of common logarithms of numbers from 1 to 10 as well as the graph of log x for values of x from 1 to 10.

x log x   x log x
1.0
0.000
2.5
0.398
1.1
0.041
3.0
0.477
1.2
0.079
3.5
0.544
1.3
0.114
4.0
0.602
1.4
0.146
4.5
0.653
1.5
0.176
5.0
0.699
1.6
0.204
6.0
0.778
1.7
0.230
7.0
0.845
1.8
0.255
8.0
0.903
1.9
0.279
9.0
0.954
2.0
0.301
10.0
1.000

Figure 3.2.1: Graph of log x

With a table of common logarithms of numbers from 1 to 10, it is easy to compute logarithms of arbitrary positive numbers simply by adding the appropriate exponent, which is the same exponent as used in scientific notation. We give a few examples below.

Example 1: Using the above table, determine the logarithm of the following numbers:

• (a) 35,000
• (b) 1,400,000
• (c) 45
• (d) 0.016
• (e) 0.0000000005

Solution:

• (a) Since 35,000 has five digits, the exponent to add to the logarithm is 4. Thus we have log 35,000 = 4 + log 3.5 = 4.544.
• (b) Since 1,400,000 has seven digits, the exponent to add to the logarithm is 6. Thus we have log 1,400,000 = 6 + log 1.4 = 6.146.
• (c) Since 45 has two digits, the exponent to add to the logarithm is 1. Thus we have log 45 = 1 + log 4.5 = 1.653.
• (d) Since 0.016 has one zero to the right of the decimal point, we must subtract the exponent 2 from log 1.6. The result is log 1.6 - 2 = 0.204 - 2 = -1.796.
• (e) Since 0.0000000005 has nine zeros to the right of the decimal point, we must subtract the exponent 10 from log 5. The result is log 5 - 10 = 0.699 - 10 = -9.301.