Logarithmic Functions
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Objectives:
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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 < y < infinity
Translate each of the following to
logarithmic form.
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23 = 8
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41/2 = 2
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Find the domain of:
F(x) = log2(x – 5)
G(x) = log5((1+x)/(1-x))
To graph logarithmic functions
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Graph the related exponential function.
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Reflect this graph across the y=x line
(Switch the x’s and y’s)
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Graph: y = log1/3x
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Natural logarithms and Common
Logarithms
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Natural Logarithm (ln) : loge
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Common Logarithm (log): log10
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Graph y=ln x (Reflect the graph of y=ex)
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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)
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Graph: f(x) = 3 log (x – 1). Determine
the domain, range, and vertical
asymptote.
Describe the translations on the graph
Solving Logarithmic Equations
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Logarithm on one side:
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Write equation in exponential form and
solve
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Examples:
Solve: log3(4x – 7) = 2
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Solve: log2(2x + 1) = 3
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Example
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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
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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
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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
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Loga1 = 0
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Logaa = 1
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alogaM = M
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Logaar = r
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Loga(MN) = logaM + logaN
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Loga(M/N) = logaM – logaN
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LogaMr = r logaM
Look at Examples Page 444-445
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Other examples:
Page 449: #8, 12, 16, 20, 24, 28, 32,
36, 44, 52, 60
Change of Base Formula:
logaM= logbM / logba
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Example: log589
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Example: log632
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Page 449: #65, 71, 74
Solving logarithmic equations
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With logarithms on both sides.
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Combine each side to one logarithm
Cancel the logarithms out
Solve the remaining equation
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Examples: Page 450: #81, 87
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Logarithm on One side of
Equation
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Combine terms into one logarithm
Write in exponential form
Solve equation that will form
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Ex: Page 454 #33, 37
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Solving Exponential Equations
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Variable is in the exponent.
Use logarithms to bring exponent down
and solve.
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Solve: 4x – 2x – 12 = 0
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Solve: 2x = 5
Solve:
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5x-2 = 33x+2
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log3x + log38 = -2
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8 .3 x = 5
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log3x + log4x = 4
Applications
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Simple Interest: I = Prt
Interest = Principal X Rate X time
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Compount Interest: A = P . (1 + r/n)nt
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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
rt
 A = Pe
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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
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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
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P = A( 1 + r/n)-nt
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P = Ae-rt
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Examples: Page 462 #5, 11, 15, 21
Exponential Decay
P = Ae-rt
Page 472 #3
Other Applications
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A(t) = Aoekt
: Exponential Growth
Newton’s Law of Cooling:
U(t) = T + (uo – T)ekt, k < 0
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Logistic Growth Model:
P(t) = c / (1 + ae-bt)
c: carrying
capacity
Examples
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Page 472: #1, 13, 22
Assignment
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Page 454, 462, 472
Exponential and Logarithmic
Regressions
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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)
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Examples
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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 = .0903e.2915x
n(7) = .0903e(.2915 x 7)
.0903e(.2915(t)) = .75
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3. b. EXP REG: y = 100.3263(.8769)x
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c. 100.3263(eln.8769)x
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d. Graph: y = 100.3263e(-.1314)x
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e. 100.3263e(-.1314)x = .5 (100.3263)
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f. 100.3263e(.1314)(50) = .141
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g. 100.3263e(-.1314)x = 20
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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)
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c.
d.
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)
Assignment
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Pages: 472, 479