Math Help: Solving for Exponents

How to Solve for Exponents

When the Bases Are Equal

In an expression like 4^8, the four is called the base, and the eight may be referred to as the exponent, power or index. As early as 8th grade, you might have to solve equations like this one: 4^x = 16. This equation is asking, 'Four to which power equals 16?' Since you know that the square root of 16 = 4, both sides of the following equation have equal bases (and both expressions equal 16): 4^x = 4^2. So, x = 2.

If you're confronted with an equation that you can't solve in your head, like 4^x = 1,024, try one of these strategies:

1. If you can't use a calculator, you'll have to estimate the value of x and manually check your answer. In this case, you'll find that 4 x 4 x 4 x 4 x 4 = 1,024, so x = 5.

2. If you can use a calculator, try inputting the number (1,024) followed by the 'x root' button and an estimate of the value of x. Your goal is to take a root of 1,024 so that the answer is the desired base (four). Since (5th root of 1,024) = 4, this means 4^5 = 1,024 and x = 5.

Without a Common Base

Although you're unlikely to encounter this type of problem in 8th grade, you may need to use logarithms to solve for exponents when you're studying more advanced algebra in high school. For instance, you may see a problem like this: 3^x = 17. Since you can't take any root of 17 and get an integer for your answer, you must undo the exponent by taking a logarithm of both sides of the equation. We'll rely on the following logarithm rule:

log-base b(x^n) = n * log-base b(x)

This rule allows us to turn the exponent into a factor, which we can then isolate using division. In the following example, we'll use the natural log (e), since it appears on most calculators. Remember that 'log-base e' is represented by 'ln.' You will need to use this function on your calculator to arrive at the last step.

3^x = 17

ln(3^x) = ln(17)

x*ln(3) = ln(17)

x*ln(3)/ln(3) = ln(17)/ln(3)

x = ln(17)/ln(3)

x = 2.5789

You will have to report an approximate value for x. You can check your answer by substituting it for x in the original equation. In this case, 3^2.5789 = 16.9999, which is approximately 17. This means that our estimate is very close to the exact value. Teachers will typically ask you to round your answer off to a specific place value for this type of problem.

Tip: You can use your calculator's x^y button to check your answer. Input the base value followed by x^y and the exponent value.