Lesson

We previously saw that a logarithm is the index that an exponential is raised to in order to get a particular result. In other words, logarithms are the inverse functions of exponentials, and exponentials are the inverse functions of logarithms.

This gives us a method to solve any equation with exponentials or logarithms. Following the general method for solving equations we perform the reverse operation to isolate the variable. When the variable is inside an exponential function we take the logarithm of the exponential, and when the variable is in a logarithmic function we take the exponential of the logarithm.

Solve \log_{64} x = \dfrac{1}{3} for x.

Worked Solution

Solve 3 \left( 10^{x}\right)= 6 for x.

Worked Solution

Solve \log_{4} 5x = 3 for x.

Worked Solution

Idea summary

Exponentials and logarithms of the same base are inverse functions of one another.

This fact allows us to solve equations involving exponentials and logarithms by applying the inverse function to both sides in order to isolate the variable.