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**7-5 Logarithmic & Exponential Equations**

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**Terms and Concepts these will be on a quiz**

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**Write original equation.**

EXAMPLE 1 Solve by equating exponents Solve 4 = x 1 2 x – 3 SOLUTION 4 = x 1 2 x – 3 Write original equation. Rewrite 4 and as powers with base 2. 1 2 (2 ) = (2 ) 2 x – 3 x – 1 2 = 2 2x – x + 3 Power of a power property 2x = –x + 3 Property of equality for exponential equations x = 1 Solve for x. The solution is 1. ANSWER

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**GUIDED PRACTICE for Example 1 Solve the equation. 3. 81 = 1 3**

= 3 – x 1 3 5x – 6 = 27 2x x – 1 SOLUTION –3 SOLUTION –6 = 1000 7x + 1 3x – 2 – 8 5 SOLUTION

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**Write original equation.**

EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x SOLUTION 4 = 11 x Write original equation. log 4x = log 4 11 4 Take log of each side. 4 log 4 x = log b = x b x x = log 11 log 4 Change-of-base formula x Use a calculator. The solution is about Check this in the original equation. ANSWER

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**GUIDED PRACTICE for Examples 2 and 3 Solve the equation. 4. 2 = 5**

= 5 x SOLUTION about 2.32 = 15 9x SOLUTION about

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**Terms and Concepts these will be on a quiz**

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**Solve a logarithmic equation**

EXAMPLE 4 Solve a logarithmic equation Solve log (4x – 7) = log (x + 5). 5 SOLUTION log (4x – 7) = log (x + 5). 5 Write original equation. 4x – 7 = x + 5 Property of equality for logarithmic equations 3x – 7 = 5 Subtract x from each side. 3x = 12 Add 7 to each side. x = 4 Divide each side by 3. The solution is 4. ANSWER

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**Exponentiate each side of an equation**

EXAMPLE 5 Exponentiate each side of an equation Solve (5x – 1)= 3 log 4 SOLUTION (5x – 1)= (5x – 1)= 3 log 4 Write original equation. 4log4(5x – 1) = 4 3 Exponentiate each side using base 4. b = x log b x 5x – 1 = 64 5x = 65 Add 1 to each side. x = 13 Divide each side by 5. The solution is 13. ANSWER

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**Solve the equation. Check for extraneous solutions.**

GUIDED PRACTICE for Examples 4, 5 and 6 Solve the equation. Check for extraneous solutions. 7. ln (7x – 4) = ln (2x + 11) 9. log 5x + log (x – 1) = 2 SOLUTION 5 SOLUTION 3 8. log (x – 6) = 5 2 log (x + 12) + log x =3 4 SOLUTION 4 SOLUTION 38

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