Remember that exponential functions and logarithmic functions

are inverses of each other.

3log 3x x3log3 x xWe will use this property to solve problems.

SOLVING EXPONENTIAL EQUATIONS

53 3x

x = 5

53 3log 3 log 3x

Solving by Equating Exponents

Solve 43x = 8 x + 1.

( 22)3x = ( 23) x + 1

6x = 3x + 3

26x = 23x + 3

22

(3x) = 23(x + 1)

x = 1

The solution is 1.

SOLUTION

Rewrite each power with base 2 .

Power of a power property

Equate exponents.

Solve for x.

4 3x = 8

x + 1 Write original equation.

CHECK Check the solution by substituting it into the original equation.

4

3 • 1 = 8

1 +

1

64 = 64 Solution checks.

Solve for x.

Solving by Equating Exponents

When it is not convenient to write each side of an exponential equation using the same base, you cansolve the equation by taking a logarithm of each side.

Taking a Logarithm of Each Side

Solve 10 2

x – 3 + 4 = 21.

10 2

x – 3 = 17

log 10 2

x – 3 = log 17

2 x = 3 + 1.23

x = (3 + 1.23 )12

x 2.115 Use a calculator.

10 2

x – 3 + 4 = 21

SOLUTION

Write original equation.

Subtract 4 from each side.

Add 3 to each side.

Multiply each side by . 12

Take log base 10 of each side.

2 x – 3 = log 17 log 10 x = x

CHECK

Taking a Logarithm of Each Side

Check the solution algebraically by substituting into theoriginal equation.

Solve 10 2

x – 3 + 4 = 21.

SOLVING LOGARITHMIC EQUATIONS

To solve a logarithmic equation, use thisproperty for logarithms with the same base:

For positive numbers b, x, and y where b 1,

log b x = log b y if and only if x = y.

Use property for logarithms with the same base.

5 x = x + 8

Solving a Logarithmic Equation

Solve log 3 (5 x – 1) = log 3 (x + 7) .

5 x – 1 = x + 7

x = 2

The solution is 2.

SOLUTION

Use inverse property with base 3.

Add 1 to each side.

Solve for x.

3 3log (5 1) log ( 7)3 3x x

Use property for logarithms with the same base.

5 x = x + 8

Solving a Logarithmic Equation

5 x – 1 = x + 7

x = 2

SOLUTION

log 3 (5 x – 1) = log 3 (x + 7) Write original equation.

Add 1 to each side.

Solve for x.

CHECK Check the solution by substituting it into the original equation.

log 3 (5 x – 1) = log 3 (x + 7)

log 3 9 = log 3 9 Solution checks.

log 3 (5 · 2 – 1) = log 3 (2 + 7)?

Write original equation.

Substitute 2 for x.

log 5 (3x + 1) = 2

Solving a Logarithmic Equation

Solve log 5 (3x + 1) = 2 .

3x + 1 = 25

x = 8

The solution is 8.

Write original equation.

Exponentiate each side

Solve for x.

Simplify.

5log (3 1) 25 5x

log 5 (3x + 1) = 2

Solving a Logarithmic Equation

Solve log 5 (3x + 1) = 2 .

5 = 5 2log5

(3x + 1)

3x + 1 = 25

x = 8

The solution is 8.

SOLUTION

Write original equation.

Exponentiate each side using base 5.

b = xlog b x

Solve for x.

log 5 (3x + 1) = 2

log 5 (3 · 8 + 1) = 2?

log 5 25 = 2?

2 = 2 Solution checks.

CHECK Check the solution by substituting it into the original equation.

Simplify.

Substitute 8 for x.

Write original equation.

Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. You can do this algebraically or graphically.

Checking for Extraneous Solutions

Solve log 5 x + log (x – 1) = 2 . Check for extraneoussolutions.

log [ 5 x (x – 1)] = 2

5 x

2 – 5 x = 100

x

2 – x – 20 = 0

(x – 5 )(x + 4) = 0

x = 5 or x = – 4

SOLUTION

log 5 x + log (x – 1) = 2 Write original equation.

Product property of logarithms.

Exponentiate both sides.

Write in standard form.

Factor.

Zero product property

Checking for Extraneous Solutions

Simplify

log(5 ( 1)) 210 10x x

The solutions appear to be 5 and – 4. However, when you check these in the original equation you can see that x = 5 is the only solution.

SOLUTION

log 5 x + log (x – 1) = 2 Check for extraneous solutions.

x = 5 or x = – 4 Zero product property

Checking for Extraneous Solutions

Check:

log 5(5) + log(5 – 1) = 2 log 5(-4) + log(-4 – 1) = 2

log 25 + log 4 = 2 log -20 + log -5 = 2

log (25)(4) = 2 error

log 100 = 2

2 = 2

If necessary use the properties of logarithms to condense several terms into one.

If necessary use the properties of logarithms to condense several terms into one.

SOLVING LOGARITHMIC EQUATIONS

Make sure that there is one term on each side of equation. Make sure that there is one term on each side of equation.

Always use the inverse property.Always use the inverse property.