Logarithmic Functions n Objectives n n Change Exponential

Logarithmic Functions n Objectives: n n 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: If ax = y, then logay = x Domain: 0 < x < infinity Range: neg. infinity < infinity

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

To graph logarithmic functions n Graph the related exponential function. n Reflect this graph across the y=x line (Switch the x’s and y’s) n Graph: y = log 1/3 x n

Natural logarithms and Common Logarithms n Natural Logarithm (ln) : loge n Common Logarithm (log): log 10 n Graph y=ln x (Reflect the graph of y=ex) n Graph y = -ln (x + 2), Determine the domain, range, and vertical asymptote. Describe the translations.

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

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

Example n 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=760 e-0. 145 h Find the height of an aircraft if the atmospheric pressure is 320 millimeters of mercury.

Example 2 n The loudness L(x), measure in decibels, of a sound of intensity x, measure in watts per square meter, is defined as L(x)=10 log(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.

Example 3 n Richter Scale: M(x) = log (x/xo) where x 0=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.

Properties of Logarithms n Loga 1 = 0 n Logaa = 1 n aloga. M = M n Logaar = r

n Loga(MN) = loga. M + loga. N n Loga(M/N) = loga. M – loga. N n Loga. Mr = r loga. M

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

Change of Base Formula: loga. M= logb. M / logba n Example: log 589 n Example: log 632 n Page 449: #65, 71, 74

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

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

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

Solve: n 5 x-2 = 33 x+2 n log 3 x + log 38 = -2 n 8. 3 x = 5 n log 3 x + log 4 x = 4

Applications n Simple Interest: I = Prt Interest = Principal X Rate X time n Compount Interest: A = P. (1 + r/n)nt n n n n 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

Compound Continuously Interest n A = Pert n 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

Finding Effective Rate of Interest n n n 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?

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

Exponential Decay P = Ae-rt Page 472 #3

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

Examples n Page 472: #1, 13, 22

Assignment n Page 454, 462, 472

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

Examples n n n Page 479: #1, 3, 7, 11 1. c. d. e. f. b. EXP REG: y =. 0903(1. 3384)x y=. . 0903(eln(1. 3384))x Graph: y =. 0903 e. 2915 x n(7) =. 0903 e(. 2915 x 7). 0903 e(. 2915(t)) =. 75

n 3. b. EXP REG: y = 100. 3263(. 8769)x n c. 100. 3263(eln. 8769)x n d. Graph: y = 100. 3263 e(-. 1314)x n e. 100. 3263 e(-. 1314)x =. 5 (100. 3263) n f. 100. 3263 e(. 1314)(50) =. 141 n g. 100. 3263 e(-. 1314)x = 20

n n n 7. b. Ln. Reg: y = 32741. 02 – 6070. 96 lnx c. Graph d. 1650 = 32741. 02 – 6070. 96 lnx = 168 computers

11. b. LOGISTIC REG (not all calculators have): Y = 14471245. 24 / (1 + 2. 01527 e-. 2458 x) n c. d. Graph Y = 14, 471, 245. 24 / (1 + 2. 01527 e-. 2458 x) Y = 14, 471, 245. 24 / (1 + 0) e. 12. 750, 854 = 14, 471, 245. 24 / (1 + 2. 01527 e -. 2458 x)

Assignment n Pages: 472, 479