Objectives Solve exponential and logarithmic equations and equalities. Solve problems involving exponential and logarithmic equations.

An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations: Try writing them so that the bases are all the same. Take the logarithm of both sides. When you use a rounded number in a check, the result will not be exact, but it should be reasonable. Helpful Hint

Solve and check. 98 – x = 27x – 3 (32)8 – x = (33)x – 3 Rewrite each side with the same base; 9 and 27 are powers of 3. 316 – 2x = 33x – 9 To raise a power to a power, multiply exponents. 16 – 2x = 3x – 9 Bases are the same, so the exponents must be equal. x = 5 Solve for x.

Check 98 – x = 27x – 3 98 – 5 275 – 3 93 272 729 729  The solution is x = 5.

5 is not a power of 4, so take the log of both sides. Solve and check. 4x – 1 = 5 log 4x – 1 = log 5 5 is not a power of 4, so take the log of both sides. (x – 1)log 4 = log 5 Apply the Power Property of Logarithms. x –1 = log5 log4 Divide both sides by log 4. x = 1 + ≈ 2.161 log5 log4 Check Use a calculator. The solution is x ≈ 2.161.

Solve and check. 32x = 27 Rewrite each side with the same base; 3 and 27 are powers of 3. (3)2x = (3)3 32x = 33 To raise a power to a power, multiply exponents. 2x = 3 Bases are the same, so the exponents must be equal. Check x = 1.5 Solve for x. 32x = 27 32(1.5) 27 33 27 27 27 

21 is not a power of 7, so take the log of both sides. Check It Out! Example 1b Solve and check. 7–x = 21 21 is not a power of 7, so take the log of both sides. log 7–x = log 21 (–x)log 7 = log 21 Apply the Power Property of Logarithms. –x = log21 log7 Divide both sides by log 7. Check x = – ≈ –1.565 log21 log7

15 is not a power of 2, so take the log of both sides. Solve and check. 23x = 15 log23x = log15 15 is not a power of 2, so take the log of both sides. (3x)log 2 = log15 Apply the Power Property of Logarithms. 3x = log15 log2 Divide both sides by log 2, then divide both sides by 3. x ≈ 1.302 Check

A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms. Review the properties of logarithms from Lesson 7-4. Remember!

Use 6 as the base for both sides. Solve. log6(2x – 1) = –1 6 log6 (2x –1) = 6–1 Use 6 as the base for both sides. 2x – 1 = 1 6 Use inverse properties to remove 6 to the log base 6. 7 12 x = Simplify.

Use 4 as the base for both sides. Solve. log4100 – log4(x + 1) = 1 100 x + 1 log4( ) = 1 Write as a quotient. 4log4 = 41 100 x + 1 ( ) Use 4 as the base for both sides. = 4 100 x + 1 Use inverse properties on the left side. x = 24

Solve. log5x 4 = 8 4log5x = 8 Power Property of Logarithms. log5x = 2 Divide both sides by 4 to isolate log5x. x = 52 Definition of a logarithm. x = 25

Product Property of Logarithms. Solve. log12x + log12(x + 1) = 1 log12 x(x + 1) = 1 Product Property of Logarithms. log12x(x +1) 12 = 121 Exponential form. x(x + 1) = 12 Use the inverse properties.

x x2 + x – 12 = 0 Multiply and collect terms. (x – 3)(x + 4) = 0 Factor. x – 3 = 0 or x + 4 = 0 Set each of the factors equal to zero. x = 3 or x = –4 Solve. Check Check both solutions in the original equation. log12x + log12(x +1) = 1 log12x + log12(x +1) = 1 log123 + log12(3 + 1) 1 x log12( –4) + log12(–4 +1) 1 log123 + log124 1 log12( –4) is undefined. log1212 1 1 1  The solution is x = 3.

Solve. 3 = log 8 + 3log x 3 = log 8 + 3log x 3 = log 8 + log x3 Power Property of Logarithms. 3 = log (8x3) Product Property of Logarithms. 103 = 10log (8x3) Use 10 as the base for both sides. 1000 = 8x3 Use inverse properties on the right side. 125 = x3 5 = x

Use 10 as the base for both sides. Solve. 2log x – log 4 = 0 2log( ) = 0 x 4 Write as a quotient. 2(10log ) = 100 x 4 Use 10 as the base for both sides. 2( ) = 1 x 4 Use inverse properties on the left side. x = 2

Lesson Quiz: Part I Solve. x = 5 3 1. 43x–1 = 8x+1 2. 32x–1 = 20 x ≈ 1.86 3. log7(5x + 3) = 3 x = 68 4. log(3x + 1) – log 4 = 2 x = 133 5. log4(x – 1) + log4(3x – 1) = 2 x = 3