In this explainer, we will learn how to solve logarithmic equations involving logarithms with different bases.
We begin by recapping the rules associated with changing the base of a logarithm.
Change of Base of Logarithms
Suppose that and are positive numbers, with and . Then,
Notice that the condition on and is just so that the logarithms are defined at all (not possible with base 1).
This result is not completely expected because what logarithms “do” is convert between products and sums, which allows for the simplification of expressions like and . Note, of course, that the change of base does NOT help with the product .
One way of understanding the terminology is that, in statement (2), in computing the logarithm of the number , we have changed the base from (in ) to the base (of the right-hand side ). The result says that in order to do this, we can just multiply by .
Recall where this comes from: the result for exponents that says that
Using it we have giving as a power of in two ways, which must be the same: .
The change of base allows us to evaluate as a logarithm to the base , since so
Although we could use the fact we have just established above, we will solve the following problem directly.
Example 1: Solving Logarithmic Equations by Changing the Base
Find the solution set of in .
We will use the fact that and change base from 9 to 3 on the right
Now, since we can recover both sides as a power of 3, we find
The solution is extraneous, because is not defined, so we conclude that the solution set .
The following example requires a little search for the application of change of base.
Example 2: Solving Logarithmic Equations Where the Unknown Is in the Base
Determine the solution set of the equation in .
Base change, assuming that a solution (where ) exists, says that so that, if we set the equation becomes or
The solution set is just .