How to Determine an Unknown Exponent


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.

Things You'll Need

  • Calculator
  • 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.

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