Section 3.4

Exponential and Logarithmic Equations

Overview

• In this section we will solve logarithmic and exponential equations.

• Some things we will need:1. The ability to convert from exponential to logarithmic

and vice versa.2. The properties of logarithms.3. The change of base formula.4. The ability to solve linear and quadratic equations5. Knowledge of the domain of a logarithmic function.

A Couple of (New) Things

1. The exponential function is one-to-one:If ax = ay, then x = y.

2. The logarithmic function is one-to-one. In both directions:If x = y, then logax = logay.

If logax = logay, then x = y.

Case I: Exponential

• Exponential equations do not have the word “log” anywhere in the problem.

• To solve an exponential equation:1. Write both sides of the equation as powers of the

same base. Then set the exponents equal to each other.

2. Take the natural log of both sides (Know the difference between an exact answer and an approximate answer).

Expressing Your Answers

• An exact answer will leave the log expressions intact. No decimal approximations will be used for any log expressions.

• An approximate answer will involve using a scientific calculator to find approximate values for log expressions. Answers will be rounded to a designated number of decimal places.

Examples

314

2

4

7

12

56

1

55

5128

xx

x

x

x

ee

Case II: Log = number

• These equations will have a log expression on one side and a number on the other.

• Solve by converting to exponential form:logax = y is the same as x = ay

Examples

182ln6

8ln

4log3

x

x

x

Case III: Multiple logs = number

• These equations will have more than one log expression on one side and a number on the other side.

• Use the properties of logarithms to combine the multiple logs into a single logarithmic expression.

• Then convert to exponential.

Case IV: Logs on both sides

• If you have multiple logs on either side, use the properties of logarithms to condense them into single logarithmic expression.

• Then set the arguments equal to each other and solve.

Examples

576log9loglog2

125loglog3

6loglog6log

5)1(log)3(log 22

x

x

xx

xx