How to Determine an Unknown Exponent

By Carter McBride
Solve natural logs easily with a calculator rather than by using long-hand math.

To solve an equation for the exponent, use natural logs in order to solve the equation. Sometimes, you can perform the calculation in your head for a simple equation, such as 4 ^ X = 16. More complicated equations require the use of algebra.

Set both sides of the equation to the natural logs. For the equation 3 ^ X = 81, rewrite as ln(3 ^ X) = ln(81).

Move X to the outside of the equation. In the example, the equation is now X ln(3) = ln(81).

Divide both sides of the equation by the logarithm on the side containing X. In the example, the equation is now X = ln(81) / ln(3).

Solve the two natural logs using your calculator. In the example, ln(81) = 4.394449155, and ln(3) = 1.098612289. The equation is now X = 4.394449155/1.098612289.

Divide the results. In the example, 4.394449155 divided by 1.098612289 equals 4. The equation, solved, is 3 ^ 4 = 81, and the value of the unknown exponent X is 4.

About the Author

Carter McBride started writing in 2007 with CMBA's IP section. He has written for Bureau of National Affairs, Inc and various websites. He received a CALI Award for The Actual Impact of MasterCard's Initial Public Offering in 2008. McBride is an attorney with a Juris Doctor from Case Western Reserve University and a Master of Science in accounting from the University of Connecticut.