In this explainer, we will learn how to solve logarithmic equations with like bases using the laws of exponents and logarithms.
Since the logarithm function to the base is inverse to the exponential function , we will also use exponential functions to solve such equations. In essence, by the definition of , it follows that given the number to solve we must have
The first steps are usually to simplify the equations using the laws of logarithms. For example, so solve it is not clear what is, so we recall that the sum of logarithms to a fixed base is just the log of a product, and rewrite the left as so that, exponentiating both sides to the 6th power, and
Example 1: Solving Logarithmic Equations
Find such that .
Since means , the statement that is the same as
We must sometimes be wary of extraneous solutions.
Example 2: Solving Logarithmic Equations Where the Unknown Is in the Base of the Logarithm
What is the solution set of the equation ?
Here, the unknown is the base. Again, we rewrite this as an exponential equation which becomes a quadratic: such that the solution set is apparently .
But we are expecting to be positive since it is a logarithmic base. This is not true for , which we dismiss as an extraneous solution. Therefore, the solution set is 6.
Example 3: Solving Logarithmic Equations
Find the solution set of in .
This looks awkward because of the squared terms: there is no simplification of either or . So, we consider this all as an equation in , from which we get so This equation of squares is solved by Rewriting these two equations using once more, implies that or implies that The solution set is .