Sometimes we know the value of an exponential expression but we don’t know the exponent that produces that value.

For example, suppose the population of a town was 1 thousand. Since then, the population has doubled every decade and is currently at 32 thousand. How many decades has it been since the population was 1 thousand?

If we say that  is the number of decades since the population was 1 thousand, then , or just , represents the population, in thousands, after  decades. To answer the question, we need to find the exponent in . We can reason that since , it has been 5 decades since the population was 1 thousand people.

When did the town have 250 people? Assuming that the doubling started before the population was measured to be 1 thousand, we can write: , or . We know that , so the exponent has a value of -2. The population was 250 two decades before it was 1,000.

But it may not always be so straightforward to calculate. For example, it is harder to tell the value of in or in . In upcoming lessons, we’ll learn more ways to find unknown exponents.