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).
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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.