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Solving Exponential and Logarithmic Equations (11.4)

Solving Exponential and Logarithmic Equations (11.4). How do we evaluate logarithms when it’s not a base 10 (log) or base e (ln)?. PROPERTY 8: The Base-Change Formula. where b can be any legitimate base. Example:. ?. ?. 4 = 4. . PROOF:. Take log b both sides:. Property 7 :.

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Solving Exponential and Logarithmic Equations (11.4)

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  1. Solving Exponential and Logarithmic Equations (11.4)

  2. How do we evaluate logarithms when it’s not a base 10 (log) or base e (ln)?

  3. PROPERTY 8: The Base-Change Formula where b can be any legitimate base Example: ? ? 4 = 4 

  4. PROOF: Take logb both sides: Property 7:  by logba: Substitute in for x:

  5. Example: Evaluate: a) log5 36 b) log½ .007 CHECK: CHECK: (½)7.1584 .007 52.2266 36.002 NOTE: You can use either bases: 10 or e. We want to use these b/c they’re in our calculators. 

  6. I. Solving Exponential Equations Example: Solve 5x = 12 for x. Hint: “x is the power of 5 that yields 12” x = log5 12 • Write as a log:  1.5440 • Use Base-Change Formula: CHECK: 51.5440 12

  7. Example: Solve 4e3x = 100 for x. •  both sides by 4: •  both sides by 4: e3x = 25 e3x = 25 • Take ln of both sides: • by definition: ln e3x = ln 25 “3x is the power of e that yields 25” • Property 7: 3x = loge25 3xln e = ln 25 3x = ln 25 • ln e = 1: 3x = ln 25  1.0730 “Notice that Base-Change Formula was not needed for this problem.”  1.0730

  8. II. Solving Logarithmic Equations NOTE: We can find logs only of (+) numbers and the base must be (+) and  1 Example: Solve log4x + log4(x + 6) = 2 for x. log4 [x(x + 6)] = 2 • Use Property 5: “2 is the power of 4 that yields x(x + 6)” 42 = x(x + 6) • Change to exp. form: 0 = x2 + 6x - 16 • Simplify & Solve: 0 = (x + 8)(x – 2) x = -8, x = 2

  9. III. Misc. Problems Example: Solve for x: • Simplify: • Notice that bases are not the same; take log4 of both sides: = 1 Use Quadratic Formula: • Property 7: x  2.550 x  -.550 • Base-Change:

  10. Example: Solve for x: Alternate Way … • Simplify: “(x2 – 2x) is the power of 4 that yields 7” • Change to exp. form:

  11. III. Misc. Problems Example: Solve for x: = 1 Use Quadratic Formula: That takes you here x  2.550 x  -.550

  12. Example: Solve for x: (log x)2 + log x3 – 12 = 0 • Use exponent property: (log x)2 + 3log x – 12 = 0 • Use z substitution: Let z = log x z2 + 3z – 12 = 0 • Use quadratic formula: z = -5.28 z = 2.28 • Substitute log x back in for z: log x = -5.28 log x = 2.28 x = 10-5.28 5.25x10-6 x = 102.28 190.55

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