A logarithmic equation has a variable inside a logarithm. Solve it by condensing to a single logarithm with the log rules, rewriting log base b of x equals y as b to the y equals x, solving the result, and checking that every argument is positive. Includes a worked example.
What a logarithmic equation is
A logarithmic equation is an equation that has a variable inside a logarithm, such as log₂(x) = 3. To solve one you undo the logarithm and find the value of the variable. It helps to be comfortable with what a logarithm is before you start.
How to solve logarithmic equations
Most logarithmic equations follow the same four steps.
Solving a logarithmic equation: condense, rewrite in exponential form, solve, then check.
The key move is step 2: converting from logarithmic to exponential form. Because logb(x) = y means exactly the same thing as by = x, rewriting the equation this way removes the logarithm and leaves an ordinary equation to solve.
Condensing with log rules
When an equation has more than one logarithm on a side, first combine them into one using the laws of logarithms — for example the product rule of logarithms, which turns log(a) + log(b) into log(ab). Once each side is a single logarithm (or a logarithm equal to a number), you can rewrite in exponential form.
Checking the answer
Every logarithm requires a positive argument, so always substitute your answer back in. Any solution that makes the inside of a logarithm zero or negative is extraneous and must be rejected.
Worked example
Solve log₂(x) = 3. Rewrite in exponential form: 2³ = x, so x = 8. Check: the argument 8 is positive, so x = 8 is valid.