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:If ax = y, then logay = x

Domain: 0 < x < infinityRange: neg. infinity < y < infinity

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))

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

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.

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

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

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.

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.

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.

Properties of Logarithms

Loga1 = 0

Logaa = 1

alogaM = M

Logaar = r

Loga(MN) = logaM + logaN

Loga(M/N) = logaM – logaN

LogaMr = r logaM

Look at Examples Page 444-445

Other examples: Page 449: #8, 12, 16, 20, 24, 28,

32, 36, 44, 52, 60

Change of Base Formula: logaM= logbM / logba

Example: log589

Example: log632

Page 449: #65, 71, 74

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

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

Solving Exponential Equations

Variable is in the exponent. Use logarithms to bring exponent

down and solve.

Solve: 4x – 2x – 12 = 0

Solve: 2x = 5

Solve:

5x-2 = 33x+2

log3x + log38 = -2

8.3x = 5

log3x + log4x = 4

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

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

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?

Present Value Formula for compounded continuously

interest

P = A( 1 + r/n)-nt

P = Ae-rt

Examples: Page 462 #5, 11, 15, 21

Exponential Decay

P = Ae-rt

Page 472 #3

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

Examples

Page 472: #1, 13, 22

Assignment

Page 454, 462, 472

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 Regressiony=c/(1+ae-bx)

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

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

7. b. LnReg: y = 32741.02 – 6070.96lnx

c. Graph

d. 1650 = 32741.02 – 6070.96 lnx= 168 computers

11. b. LOGISTIC REG (not all calculators have):

Y = 14471245.24 / (1 + 2.01527e-.2458x)

c. Graph

d. 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)

Assignment

Pages: 472, 479