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Solving for a Variable in an Exponent

Sometimes algebra problems will put the variable that you have to solve for in an exponent.  These problems can look intimidating but work exactly the same as other “plug in” problems.  Just find the number that makes the exponent equation true!

Getting the vocabulary right makes it easier to talk about exponent problems so:

  • Exponent: The small “superscript” number that multiplies another number by a power (which means it multiplies that number by itself a specified number of times). In 63 the little “3” is the exponent.
  • Base: The larger (in size, not necessarily in value) number that is being affected by the exponenet.   In 63 the 6 is the base.

Even though we're used to seeing exponents as a single number, they can also be little equations.

The best way to solve these questions is to figure out what the exponent should be and then solve the little exponent equation for the variable.

Example: $5^x=25$, solve for $x$.

The first thing you need to do is figure out $5^\text{?}=25$

$5^2=25$, so $x=2$.

What about when there is a little equation in the exponent?

Example: $4^{x+1}=64$, solve for $x$.

Again, the first thing you need to do is find $x^?=64$

$$\eqalign{4\times4\times4&=64\\4^3&=64\\\text{Now create a new equation,}&\text{ setting the exponent equation equal to 3}\\x+1&=3\\-1&\;\;-1\\x&=2}$$

Overall, solving an for a variable in an exponent is the same as solving any other kind of equation.  Just figure out what the exponent should be, then set the exponent equation equal to that exponent and solve.

Practice Problems:

Test Prep Practice

Common Core Grade Level/Subject