Logarithms

Calculator keys showing log to base 10 on the left and natural log on the right

To understand logs we need to understand the work on powers that we saw in the last section. The log of a number is the power the base has to be raised to in order to give the original number. An example should make this clearer:

log₁₀1000

To get 1000 we have to raise 10 (the base) to the power 3 as 10³ = 1000

Consequently, log₁₀1000 = 3

log₁₀0.01

To get 0.01 we have to raise 10 (the base) to the power -2 as 10^-2 = 0.01

Consequently, log₁₀0.01 = -2

To reverse the process of taking a log of a number we have to raise the base to the log. In the example given above, where log₁₀ = 3, the original number must be 10³ = 1000.

Natural logs use the base e (a number approximately equal to 2.7182). This number is chosen for mathematical reasons. The principle behind natural logs is exactly the same as that for the base 10 logs.

Logs have the useful property that adding them together has the effect of multiplying together the original numbers. Consequently, logs were used before the existence of calculators for multiplying numbers together, as adding is easier than multiplying. The logs were read from books of log tables which also gave tables for reversing the log process in order to recover the result for the multiplied numbers.

ln(x) is directly proportional to log₁₀(x), the conversion factor being ln(10), which is approximately 2.303. So ln(x) = 2.303 × log₁₀(x). As all modern scientific calculators have both log₁₀ and ln functions this seems of little value, but some specifications still make use of it.