Common Logarithms
Since our number system uses base 10, logarithms to base 10 are frequently used.
Base 10 logarithms are called common logarithms and are written simply as log x.
It should come as no surprise, then, that a number written in exponential form with
base 10 can easily be converted to a common logarithm and vice versa.
Exponential Form
1000 = 103
100 = 102
10 = 101
1 = 100
1
10
1
100
Logarithmic Form
log 1000 = 3
log 100 = 2
log 10 = 1
log 1 = 0
1
log 0.1 = log ( ) = -1
10
= 0.1 = 10-1
log 0.01 = log (
= 0.01 = 10-2
1
100
) = -2
Example 1: Find the log 1,000,000
1,000,000 = 106
log 1,000,000 = 6
Write 1,000,000 in exponential form
Convert exponential form to logarithmic form
Example 2: Find the log 0.0001
0.0001 = 10-4
log 0.0001 = -4
Write 0.0001 in exponential form
Convert exponential form to logarithmic form
It is not as easy to find the common logarithm of a number that is not a power of 10.
We once used a table of logarithms, but now we use a scientific calculator.
calculator key
log
The
is used to find common logarithms. For some calculators,
you enter the number first and then press
log
while for others you press
log
first and then enter the number – it depends on the brand of your calculator.
Example 3: Use a calculator to approximate log 93 to four decimal places
log 93 ≈ 1.9685
Example 4: Use a calculator to approximate log 0.00251 to four decimal places
log 0.00251 ≈ -2.6003
Example 5: Use a calculator to approximate log 500 to four decimal places
Log 500 ≈ 2.69897