Exponential equations

In this section we will refer to

as one of the inverse properties.

and

as property 3 (since it was third on our list in the section on properties of logarithms).

Example 1

Solve

Solution.

First isolate the exponential part . To do this we add -1 to both sides. This gives us

Now take either log or ln of both sides of the equation . Which one you use is your choice but you must use the same one on both sides. Here we take ln of both sides.

Apply property 3.

Recall that property 3 says in this case that Hence our equation becomes

Dividing both sides by ln(3), yields

= = 2.680143859

As a check graph and the line y =20 and look at where they intersect.

Example 2

Solve

Solution .

First isolate the exponential part . In this case this means we must first divide both sides by 500. This gives us.

Take ln of both sides since base e is involved. This gives us

Using the inverse property on the left side, we get

Dividing by 4, yields = = .173286795

The following graph shows that this is a reasonable result.

Example 3

Solve

Solution .

Isolate the exponential part by dividing both sides by 100. This give us

Taking the natural logarithm of both sides , yields

By the inverse property we get

Thus, = 30.09932011

As a check graph and y = 30 and look at where they intersect.

Example 4

Solve

Solution .

Isolate the exponential by dividing both sides by 300.

= .25

Taking the natural log of both sides, yields

Using property 3 , we get

Dividing both sides by gives us

= .5

To confirm this we will subsitute this into the original equation. We get

= = 75

General Strategy

For the types of equations considered in this section,

we have isolated the exponential part,

taken the natural log of both sides,

used either the inverse property or property 3,

and then solved the equation.