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>> 3.3. Properties of Logarithms
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 baseb logarithm of a positive real number x, written as log_{b}x, is equal to the unique number y such that b^{y} = x. This definition works because exponential functions are onetoone, meaning that for b and x fixed, the exponent y such that b^{y} = x is unique.
The most commonly used logarithms used to be base10 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, base10 logarithms are known as common logarithms, and the subscript 10 is omitted from their notation, i.e. instead of writing log_{10}x 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 

onetrillionth  0.000000000001 
10^{12} 
12 
one hundredbillionth  0.00000000001 
10^{11} 
11 
one tenbillionth  0.0000000001 
10^{10} 
10 
one billionth  0.000000001 
10^{9} 
9 
one hundredmillionth  0.00000001 
10^{8} 
8 
one tenmillionth  0.0000001 
10^{7} 
7 
one millionth  0.000001 
10^{6} 
6 
one hundredthousandth  0.00001 
10^{5} 
5 
one tenthousandth  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 
10^{0} 
0 
ten  10 
10^{1} 
1 
one hundred  100 
10^{2} 
2 
one thousand  1,000 
10^{3} 
3 
ten thousand  10,000 
10^{4} 
4 
one hundred thousand  100,000 
10^{5} 
5 
one million  1,000,000 
10^{6} 
6 
ten million  10,000,000 
10^{7} 
7 
one hundred million  100,000,000 
10^{8} 
8 
one billion  1,000,000,000 
10^{9} 
9 
ten billion  10,000,000,000 
10^{10} 
10 
one hundred billion  100,000,000,000 
10^{11} 
11 
one trillion  1,000,000,000,000 
10^{12} 
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.
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