Solving Exponential Equations

An exponential equation is an equation that has the unknown in the exponent. We will explore two analytical methods for solving exponential equations in this section.

I. Solving Exponential Equations Using Exponential Forms

Steps:

1. Rewrite the equation with the term containing the exponent by itself on one side.

2. Divide both sides by the coefficient (multiplier) of the term containing the exponent.

3. Change the new equation to logarithmic form.

4. Solve for the variable.

 

Before we try the second analytical method, we need some properties of logarithms.

 

Basic Properties of Logarithms:

For b > 0, b ≠ 1:

1. log _b b = 1
2. log _b 1 = 0
3. log _b b^x = x
4. b ^{log b x} = x

The next three logarithmic properties are helpful in simplifying logarithmic expressions. These three properties have names.

Additional Logarithmic Properties:

Again, for b > 0, b ≠ 1 and for k a real number and M and N positive real numbers:

5. Product Property  log _b (MN) = log _b M + log _b N
6. Quotient Property
7. Power Property log _b M^k = k log _b M

 

II. Solving Exponential Equations using Logarithmic Properties

Steps:

1. Rewrite the equation with a base raised to a power on one side.

2. Take the logarithm, base e or 10, of both sides of the equation.

3. Use a logarithmic property to remove the variable from the exponent.

4. Solve for the variable.

 

Solving Logarithmic Equations

Quite often we can solve logarithmic equations by using the definition of a logarithm:

y = log _b x ↔ b^y = x. Sometimes we need to use the properties of logarithms to get the equation in the form y = log _b x first.

Note: Solutions of logarithmic equations must be checked to make sure that all arguments in the original equation remain positive.