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Mastering Logarithms: Equations and Models

Learn to solve exponential and logarithmic equations, understand properties of logarithms, and model data with exponential, logarithmic, and logistic functions. Practice using the change of base formula and apply key concepts in math.

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Mastering Logarithms: Equations and Models

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  1. 5.5 Objectives • Apply the base properties of logarithms. • Use the change of base formula.

  2. Properties of Logarithms • loga1=0 and logaa=1 • logam+logan=loga(mn) • logam-logan=loga(m/n) • loga(mr)=rlogam ln = loge (natural log)

  3. Change of base formula • Let x, a, and b be positive numbers.. • ..where a≠ 1 and b≠ 1.

  4. Try these • ln 5+ ln 4 log 10-log 5

  5. And more… log 52 log 5 + log 15 – log 10

  6. 5.6 Objectives • Solve exponential equations • Solve logarithmic equations.

  7. Exponential Equations Basic form: Cax = k • Solve for ax • take the base a log of both sides, which makes the ax equal x because.. logaax = x • Solve the other side

  8. Try these Log (2x+1) =2

  9. Try these Log2 4x = 2-log2 x

  10. Logarithmic equations • Basic form: C logax = k. • Solve for logax. • Exponentiate each side with base a. This makes the logax side equal x because alogax= x • Solve.

  11. Try these log x + log (2x+1) = log 7 2log23x= 1

  12. 5.7 Objectives • Find an exponential model. • Find a logarithmic model.

  13. Types of models • Exponential • f(x) = Cax • Can be used to model data that increase or decrease rapidly over time • Logarithmic • f(x) = a + b logx • Can be used to model data that increase gradually over time • Logistic • f(x) = • Can be used to model data that at first increase slowly, then increase rapidly, and finally level of

  14. Exponential model f(x) = Cax

  15. Logarithmic model f(x) = a + b logx

  16. Logistic model

  17. assignment • Page 442-443 • 7-14 • 31-38 • Page 453 • 5-14 • 33-38 • Page 462 • 1-4

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