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