Solving Exponential Equations using Logarithms

1. Review of Properties of Logs

Exponential equations can be equivalently written using a logarithm. In general, we can say that the following two equations are equivalent:

	Exponential equation
	Logarithmic equation

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 of logs and the change of base property of logs:

formula to know

Power Property of Logs

Change of Base Property of Logs

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).

2. Writing Exponential Equations as Logarithms

EXAMPLE

Consider the equation: . Solve for x.

One strategy to solve 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:

	Exponential equation
	Logarithmic 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:

	The logarithmic equation.
	Use the change of base property.
	Use a calculator to find the solution of 1.7 (rounded to the nearest tenth).

So, in our expression , is our solution.

3. 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 and other inverse operations to isolate the variable.

EXAMPLE

Find the solution for x in the equation .

	The exponential equation.
	To solve the equation , begin by dividing both sides by 3 to cancel the coefficient in front of the exponent.
	To undo the variable exponent, apply a logarithm to both sides. This will allow us to move the variable exponent outside the parentheses and isolate the variable.
	Next, apply the power property of logs, which states that exponents inside a logarithm can be expressed as outside scalar multiples of the logarithm. This means that can be expressed as .
	We can treat and as numbers, and divide both sides by to isolate x.
	To solve, you can use a calculator to evaluate and . is approximately equal to 0.4771 and is approximately equal to 0.3010.
	Finally, divide to solve for x to find the solution to the equation .

EXAMPLE

Find the solution for x in the equation .

	The exponential equation.
	Divide by 2 to have only on the left side.
	Apply the log of both sides.
	Use the power property of logs.
	Divide both sides by to isolate x.
	Use a calculator to find a solution of 2.6

IN CONTEXT

The equation shows the number of infected people from an outbreak of the norovirus. The variable y represents the number of infected people, and t represents time in weeks.

In how many weeks will the number of infected people reach 13,122?

To answer this question, we start by substituting 13,122 in for y, the number of infected people, and solve for t:

	The exponential equation.
	Substitute 13,122 in for y.
	Divide both sides by 2, the coefficient.
	If the variable that needs to be solved is in the exponent, you can apply a log to both sides of the exponential equation.
	Use the power property of logs to move the exponent, t, outside of the parentheses.
	Divide both sides by to isolate t.
	Use a calculator to evaluate the logs (you can round to the third decimal).
	Divide and round your answer to complete weeks.

It will take about 8 weeks for the number of infected people to reach 13,122.

big idea

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.