# How To Find The Base of a Logarithm?

Solving for an unknown base of a logarithm simply means working in a reverse form. In this case, you will have a logarithm of a number without knowing the base of that logarithm. From our knowledge of how to evaluate basic logarithm, this reverse process should not pose any challenge to us.

The general notation of a logarithm is

where the small letter b is the base, and the small letter x is the exponent or the power. We can convert this logarithmic equation into an exponential equation to be

From here, we can work logarithms either forward or backward, always finding the unknown in the equation.

Example 1

Given that

find the base of the log.

Solution

Next, we solve for b by taking the cube root of both sides of our new equation.

Example 2

Evaluate:

Solution

First write the equation in an exponential form.

Second, apply exponential laws:

We solve for b by taking the 4th root of both sides of the equation.

Example 3

Evaluate:

Solution

As usual, we start by rewriting the log equation into an exponential equation.

We solve for b by taking the cube root of both sides of the equation.

## Remarks

The essential knowledge you need to find the base of a logarithm is the exponential laws and a bit of algebra. From the tutorial, you should be able to evaluate the base of any logarithm from a given equation. In short: Rewrite the logarithmic equation you have into an exponential form and solve for the unknown, which is the base of the logarithm.