Intermediate Algebra (11th edition)

SECTION 10.6 Exponential and Logarithmic Equations; Further Applications 613

OBJECTIVES

Exponential and Logarithmic Equations;

Further Applications

10.6

1 Solve equations

involving variables

in the exponents.

2 Solve equations

involving

logarithms.

3 Solve applications

of compound

interest.

4 Solve applications

involving base

eexponential

growth and decay.

We solved exponential and logarithmic equations in Sections 10.2 and 10.3.General

methods for solving these equations depend on the following properties.

Properties for Solving Exponential and Logarithmic Equations

For all real numbers and any real numbers xand y, the following

are true.

1. If then

2. If then

3. If and then

4. If and then x 7 0, y 7 0, logb x=logb y, x=y.

x= y, x 7 0, y 7 0, logb x= logb y.

bx=by, x= y.

x= y, bx =by.

b 7 0,bZ1,

We used Property 2 to solve exponential equations in Section 10.2.

OBJECTIVE 1 Solve equations involving variables in the exponents.In

Examples 1 and 2,we use Property 3.

Solving an Exponential Equation

Solve Approximate the solution to three decimal places.

Property 3 (common logs)

Power rule

Exact solution Divide by log 3.

Decimal approximation Use a calculator.

CHECK ✓ Use a calculator; true

The solution set is 5 2.262 6. NOW TRY

3 x= 3 2.262L 12

xL2.262

x=

log 12

log 3

x log 3= log 12

log 3 x= log 12

3 x= 12

3 x= 12.

NOW TRY EXAMPLE 1

EXERCISE 1
Solve the equation.
Approximate the solution
to three decimal places.

5 x= 20

NOW TRY ANSWER

1. 5 1.861 6

CAUTION Be careful: is notequal to log 4. Check to see that

but

log 12

log 4L0.6021, log 3 L2.262.

log 12

log 3

When an exponential equation has eas the base, as in the next example, it is eas-

iest to use base elogarithms.