3.4 Solving Exponential and Logarithmic Equations

There are 2 main methods you can use to solve equations involving exponential and logarithmic functions:

1) The “Key to Everything”

This states that:

if and only if

This is the definition of a logarithm.

2) The One-to-One Property

This states that when

and

if and only if

This property works because exponential and logarithmic functions are one-to-one, meaning that for every x value, there is only one corresponding y value. In other words, they pass the vertical line test.

To illustrate the one-to-one property:

, we can conclude that x=3.

It is also helpful to know the Inverse Properties of logarithms:

We can show that this is true using the key to everything.

Rearranging the above equation, we find that:

Again, the key to everything helps us prove this. Rearranging the equation, we see that:

Knowing all the other properties of logarithms, which can be found in the post on section 3.3, will also be very useful in solving such equations.

Examples:

Solving Exponential Equations

1) Solve for x.

Given this equation, we can simplify the right side:

Using the one-to-one property, we can then conclude that x=-5.

2) Solve for x.

Using the key to everything, we find that:

Since a logarithm with a base of “e” is the same as a natural log, we simplify to:

The answer can be written as x=ln3, which is approximately equal to 1.099.

Solving Exponential Equations in Quadratic Form

3) Solve

First, we substitute in

to get:

Then, we factor and solve like a regular quadratic:

and

Next, we substitute

again:

and

Finally, using the key to everything, we find that:

and *

*HOWEVER: The solution x=ln(-2) is extraneous, because you cannot take the logarithm or the natural logarithm of a negative number. Therefore, the only real solution to this equation is x=ln5.

Solving Logarithmic Equations

4) Solve