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Logarithmic Functions

Logarithmic Functions. Objectives: Change Exponential Expressions <-  Logarithmic Expressions Evaluate Logarithmic Expressions Determine the domain of a logarithmic function Graph and solve logarithmic equations. Logarithmic Functions. Inverse of Exponential functions:

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Logarithmic Functions

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  1. Logarithmic Functions • Objectives: • Change Exponential Expressions <-  Logarithmic Expressions • Evaluate Logarithmic Expressions • Determine the domain of a logarithmic function • Graph and solve logarithmic equations

  2. Logarithmic Functions Inverse of Exponential functions: If ax = y, then logay = x Domain: 0 < x < infinity Range: neg. infinity < y < infinity

  3. Translate each of the following to logarithmic form. • 23 = 8 • 41/2 = 2 • Find the domain of: • F(x) = log2(x – 5) • G(x) = log5((1+x)/(1-x))

  4. To graph logarithmic functions • Graph the related exponential function. • Reflect this graph across the y=x line • (Switch the x’s and y’s) • Graph: y = log1/3x

  5. Natural logarithms and Common Logarithms • Natural Logarithm (ln) : loge • Common Logarithm (log): log10 • Graph y=ln x (Reflect the graph of y=ex) • Graph y = -ln (x + 2), Determine the domain, range, and vertical asymptote. Describe the translations.

  6. Graph: f(x) = log x (Reflect the graph of y = 10x) • Graph: f(x) = 3 log (x – 1). Determine the domain, range, and vertical asymptote. • Describe the translations on the graph

  7. Solving Logarithmic Equations • Logarithm on one side: • Write equation in exponential form and solve • Examples: • Solve: log3(4x – 7) = 2 • Solve: log2(2x + 1) = 3

  8. Example • The atmospheric pressure ‘p’ on a balloon or an aircraft decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height ‘h’ (in kilometers) above sea level by the formula p=760e-0.145h Find the height of an aircraft if the atmospheric pressure is 320 millimeters of mercury.

  9. Example 2 • The loudness L(x), measure in decibels, of a sound of intensity x, measure in watts per square meter, is defined as L(x)=10log(x/Io) where Io = 10-12 watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of heavy city traffic: intensity of x=10-3 watt per square meter.

  10. Example 3 • Richter Scale: M(x) = log (x/xo) where x0=10-3 is the reading of a zero-level earthquake the same distance from its epicenter. Determine the magnitude of the Mexico City earthquake in 1985: seismographic reading of 125,892 millimeters 100 kilometers from the center.

  11. Properties of Logarithms • Loga1 = 0 • Logaa = 1 • alogaM = M • Logaar = r

  12. Loga(MN) = logaM + logaN • Loga(M/N) = logaM – logaN • LogaMr = r logaM

  13. Look at Examples Page 444-445 • Other examples: • Page 449: #8, 12, 16, 20, 24, 28, 32, 36, 44, 52, 60

  14. Change of Base Formula: logaM= logbM / logba • Example: log589 • Example: log632 • Page 449: #65, 71, 74

  15. Solving logarithmic equations • With logarithms on both sides. • Combine each side to one logarithm • Cancel the logarithms out • Solve the remaining equation • Examples: Page 450: #81, 87

  16. Logarithm on One side of Equation • Combine terms into one logarithm • Write in exponential form • Solve equation that will form • Ex: Page 454 #33, 37

  17. Solving Exponential Equations • Variable is in the exponent. • Use logarithms to bring exponent down and solve. • Solve: 4x – 2x – 12 = 0 • Solve: 2x = 5

  18. Solve: • 5x-2 = 33x+2 • log3x + log38= -2 • 8.3x = 5 • log3x + log4x = 4

  19. Applications • Simple Interest: I = Prt • Interest = Principal X Rate X time • Compount Interest: A = P . (1 + r/n)nt • Time is in years • Annually: once a year • Semiannually: Twice per year • Quarterly: Four times per year • Monthly: 12 times per year • Daily: 365 times per year

  20. Compound Continuously Interest • A = Pert • The present value P of A dollars to be received after ‘t’ years, assuming a per annum interest rate ‘r’ compounded ‘n’ times per year, is P=A.(1 + r/n)-nt

  21. Finding Effective Rate of Interest • On January 2, 2004, $2000 is placed in an Individual Retirement Account (IRA) that will pay interest of 10% per annum compounded continuously. • What will the IRA be worth on January 1, 2024? • What is the effective rate of interest?

  22. Present Value Formula for compounded continuously interest • P = A( 1 + r/n)-nt • P = Ae-rt • Examples: Page 462 #5, 11, 15, 21

  23. Exponential Decay P = Ae-rt Page 472 #3

  24. Other Applications • A(t) = Aoekt : Exponential Growth • Newton’s Law of Cooling: U(t) = T + (uo – T)ekt,k < 0 Logistic Growth Model: P(t) = c / (1 + ae-bt) c: carrying capacity

  25. Examples • Page 472: #1, 13, 22

  26. Assignment • Page 454, 462, 472

  27. Exponential and Logarithmic Regressions • Input data into calculator • Go to calculate mode • Find ExpReg (Exponential Regression) y = abx • Find LnReg (natural logarithm regression) y = a + b.lnx • Logistic Regression y=c/(1+ae-bx)

  28. Examples • Page 479: #1, 3, 7, 11 • 1. b. EXP REG: y = .0903(1.3384)x • c. y=..0903(eln(1.3384))x • d. Graph: y = .0903e.2915x • e. n(7) = .0903e(.2915 x 7) • f. .0903e(.2915(t)) = .75

  29. 3. b. EXP REG: y = 100.3263(.8769)x • c. 100.3263(eln.8769)x • d. Graph: y = 100.3263e(-.1314)x • e. 100.3263e(-.1314)x = .5 (100.3263) • f. 100.3263e(.1314)(50) = .141 • g. 100.3263e(-.1314)x = 20

  30. 7. b. LnReg: y = 32741.02 – 6070.96lnx • c. Graph • d. 1650 = 32741.02 – 6070.96 lnx = 168 computers

  31. 11. b. LOGISTIC REG (not all calculators have): Y = 14471245.24 / (1 + 2.01527e-.2458x) • Graph • Y = 14,471,245.24 / (1 + 2.01527e-.2458x) Y = 14,471,245.24 / (1 + 0) e. 12.750,854 = 14,471,245.24 / (1 + 2.01527e-.2458x)

  32. Assignment • Pages: 472, 479

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