Exponential Equations with Logarithms: Problem Solving

In a forested region with a wide beaver habitat, a researcher carefully tracks how the beaver population grows over time.

The goal is to determine the number of years needed for the population to reach a specific size.

The population follows an exponential model based on repeated growth over time. It equals the initial population multiplied by 10 raised to the growth rate times the number of years. The growth rate shows how fast the population increases each year.

To begin the calculation, the researcher substitutes the target population value into the equation.

Dividing both sides by the initial population yields the factor by which the population has grown. The equation is then rearranged so that ten raised to an exponent equals that factor.

Since logarithms and exponents are inverse operations, taking the logarithm of both sides isolates the variable. Then, applying the power law brings the exponent down, turning the equation into a solvable linear form.

The exponent now clearly appears as a product of the constant and the number of years.

Dividing the logarithmic value by the constant gives the estimated number of years it will likely take for the population to reach the expected final population size.