# Solve for Exponents Calculator

## Calculator Use

This calculator will solve for the exponent n in the exponential equation x^{n} = y, stated x raised to the nth power equals y. Enter x and y and this calculator will solve for the exponent n using log(). Since taking the log() of negative numbers causes calculation errors they are not allowed.

## How to solve for exponents

For x^{n} = y; solve for n by taking the log of both sides of the equation:

For:

\( x^n = y \)

Take the log of both sides:

\( \log_{}x^n = \log_{}y \)

By identity we get:

\( n \cdot \log_{}x = \log_{}y \)

Dividing both sides by log x:

\( n = \dfrac{\log_{}y}{\log_{}x} \)

### Find the exponent of a number

For the equation 3^{n} = 81 where 3 is called the base and n is called the exponent, find the value of the exponent n using logarithms.

For:

\( 3^n = 81 \)

Take the log of both sides:

\( \log_{}3^n = \log_{}81 \)

By identity we get:

\( n \cdot \log_{}3 = \log_{}81 \)

Dividing both sides by log 3:

\( n = \dfrac{\log_{}81}{\log_{}3} \)

Using a calculator we can find that log 81 ≈ 1.9085 and log 3 ≈ 0.4771 then our equation becomes:

\( n = \dfrac{\log_{}81}{\log_{}3} \approx \dfrac{1.9085}{0.4771} \approx 4 \)

Checking our answer **3 ^{4} = 81**.

Since taking the log() of negative numbers, 0 or 1 causes calculation errors we have provided some answers by definition and not actual calculations.