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How to Isolate Bases and Exponents

Much like multiplication is shorthand for repeated addition, expressions with exponents are shorthand for repeated multiplication. Raising a number, called the base, to an exponent repeatedly multiplies the base number by itself a set number of times. That number is equal to the value of the exponent. For example, 3^5 = (3 * 3 * 3 * 3 * 3) = 243. A logarithm is a mathematical operation the "undoes" the work of an exponent. Logarithms are necessary when isolating bases from their exponents, such as when simplifying an exponential equation.

Instructions

Isolating the Exponent
- 1
  Take the logarithm of both sides of the equations. This brings down the exponent to the front of the base. For example, for the equation (5^x) = 25, taking the log of each side finds: (x)(log(5) = log(25).
- 2
  Divide both sides of the equation by the log of the base of the exponent you want to isolate. For example, (x)(log(5) = log(25) becomes: ((x)(log(5) / log(5)) = (log(25) / log(5)) then x = (log(25) / log(5)).
- 3
  Solve for the exponent if necessary. The exponent is now isolated from its base, though its value may still need to be determined. For example, x = (log(25) / log(5)) = 2.
Isolating the Base
- 4
  Take the root equal to the exponent of the base you want to isolate. For example, for the equation x^4 = 81, take the 4th root of both sides of the equation. The fourth root is equivalent to taking the square root twice: sqrt(sqrt(x^4)) = sqrt(sqrt(81))
- 5
  Drop the exponent from the base as the root function cancels it out. For example, sqrt(sqrt(x^4)) = sqrt(sqrt(81)) becomes x = sqrt(sqrt(81)).
- 6
  Solve for the base if necessary. The base is now isolated from the exponent, though its value may still need to be determined. For example, x = sqrt(sqrt(81)) = sqrt(9) = 3.

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