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7-5 Logarithmic & Exponential Equations

7-5 Logarithmic & Exponential Equations. Terms and Concepts these will be on a quiz. 1. x – 3. x . 2. Solve 4 = . x – 3. 2 x. – x + 3. x . 2 = 2 . 1. 1. 4 =. Rewrite 4 and as powers with base 2 . 2. 2. x. x – 3. 2. – 1. (2 ) = (2 ).

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7-5 Logarithmic & Exponential Equations

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  1. 7-5 Logarithmic & Exponential Equations

  2. Terms and Conceptsthese will be on a quiz

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

  4. 1 – 8 3 5 5x –6 3 – x 3. 81 = 2x x –1 1. 9 = 27 7x +1 3x –2 2. 100 = 1000 for Example 1 GUIDED PRACTICE Solve the equation. SOLUTION –3 SOLUTION –6 SOLUTION

  5. log 4x= log 4 11 4 Takelogof each side. 4 x = log 11 x = 11 log 4 x 4 = 11 log 4 x x Solve 4 = 11. log b = x b The solution is about 1.73. Check this in the original equation. ANSWER x 1.73 EXAMPLE 2 Take a logarithm of each side SOLUTION Write original equation. Change-of-base formula Use a calculator.

  6. 9x 5. 7 = 15 x 4. 2 = 5 for Examples 2 and 3 GUIDED PRACTICE Solve the equation. SOLUTION about 2.32 about 0.155 SOLUTION

  7. Terms and Conceptsthese will be on a quiz

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

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

  10. 10.log (x + 12) + log x =3 4 4 8.log(x – 6) = 5 2 for Examples 4, 5 and 6 GUIDED PRACTICE 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 4 SOLUTION SOLUTION 38

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