Exponential equations can be equivalently written using a logarithm. In general, we can say that the follow two equations are equivalent:
Notice that the base of the exponential expression becomes the base of the logarithmic expression. Also notice that y, which was the output of the exponential equation, is the input of the logarithmic operation. This is characteristic of inverse relationships.
When solving exponential equations using logarithms, we often apply two important properties of logarithms: the power property and the change of base property:
We most often use the change of base property when we wish to use our calculators to evaluate logs. This is because most calculators can only evaluate logs in base 10 (the common log) or e (the natural log).
Writing Exponential Equations as Logarithms
Consider the equation:
One strategy to solving this equation is to see if 10.556 is a power of 4. If so, it is relatively easy to solve for x mentally, as x will be an integer, such as 1, 2, 3, and so on. However, there is no integer exponent we can apply to 4 to get a value of 10.556. In this case, it is helpful to write this into an equivalent exponential equation:
Now that we have a logarithmic expression for x, we can use the change of base property to evaluate the log using our calculator:
Applying Logarithms to Both Sides of an Equation
As with all equations, we can apply inverse operations to both sides of the equation in order to isolate the variable. The inverse operation of an exponent is the logarithm. In this next example, we will see how we can apply the logarithm to isolate the variable.
When applying the log to both sides of an exponential equation, this enables us to apply the power property of logarithms. We can move the variable outside of the logarithm, as a scalar multiplier to the log. Then, we are able to isolate the variable by dividing everything else out.