Logarithms were introduced to help simplify complicated calculations. The common logarithm (logarithm with base 10) of a number n is the exponent r, where 10r= n. Note the similarities to the logarithmic scales discussed previously.
We write the logarithm of a number n as log(n).
For example:
log(1) = 0, since 100 = 1
log(10) = 1, since 101 = 10
log(100) = 2, since 102= 100
log(1000) = 3, since 103 = 1000
similarly, log(0.01) = -2, since 10-2 = 1/100.
The logarithm function with base 10 is r = log(n), where n is the explanatory variable and r is the response variable.
10n and log(n) are related. In fact, they are inverse functions.
Note in the graph how one can be obtained from the other by reflecting across y = n.
Also, note that log(n) can only be evaluated for n > 0, since 10r is positive for any value of r.
Properties of Logarithms
(1) log(a*b) = log(a) + log(b)
(2) log(a/b) = log(a) - log(b)
(3) log(ar) = r * log(a)
Example 7.6
log(20) = log(2*10) = log(2) + log(10) = 0.30 + 1 = 1.30
log(20/2) = log(20) - log(2) = 1.30 - 0.30 = 1
log(100) = log(102) = 2 * log(10) = 2 * 1 = 2
Scientific Notation is used to more easily express very large or very small numbers. For example, 4264 would be 4.264 x 103 in scientific notation. In most calculators and software packages, it is represented as 4.264 e 3.
Similarly, 0.00124 would be 1.24 x 10-3, or 1.24 e -3.