Logarithms
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of
More generally, if
It is useful to think of logarithms as inverses of exponentials. So, for example:
And:
Product Rule for Logarithms
Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. Logarithms were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily by using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition, because of the fact that the logarithm of a product is the sum of the logarithms of the factors:
We can see that this rule is true by writing the logarithms in terms of exponentials.
Let
Writing these equations as exponentials:
And:
Then note that:
Taking the logarithm base
This is a very useful property of logarithms, because it can sometimes simplify more complex expressions. For example:
Then because