Calculators

Algebra

This calculator will solve for the exponent n in the exponential equation xⁿ = 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 xⁿ = 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 3ⁿ = 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 3⁴ = 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.