The following two equations are equivalent:
Here, is the logarithm of to the base . In other words, needs to be raised to the power of to equal .
Logarithms are only defined when is positive and not equal to , and is positive.
Two of the most common log bases you will encounter are and ( is Euler's number). They are sometimes written as simply and , respectively.
Several important identities are used to relate logarithms to one another. They are known as logarithmic laws.
Here are a couple of example problems:
Solve for :
Use the laws of logarithms to put into a single logarithm.
This logarithmic expression can then be rewritten as an exponential.
This gives us the quadratic equation , which has the roots and .
We immediately have to reject , because and are negative and, therefore, undefined. Thus, the correct answer is
Application and Extensions
Here is an application that uses all five logarithmic laws:
Rewrite as a single logarithm: