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The student will learn about:

the logarithmic function,

its properties, and

applications.

§3.3 Logarithmic Functions.

Introduction

• In this section we introduce logarithmic functions, emphasizing the natural logarithm function.

• We then apply natural logarithms to a wide variety of problems, from doubling money under compound interest to carbon 14 dating.

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The Logarithmic Function

Definition.  The inverse of an exponential function is called a logarithmic function. That is, where b > 0, and b ≠ 1;

if y = log b x , then b y = x .

The domain of the log function is the set of all positive real numbers and the range of the log function is the set of all positive real numbers.

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Examples

Log 3 9 = 2

Log 2 16 = y, find y.

Log 5 x = 3, find x.

Log b 1/8 = -3, find b.

So, y = 4. y2 1635 x

3b 1 / 8

So, x = 125.

So, b = 2.

Since 3 2 = 9

If y = log b x , then b y = x

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Calculator Evaluation of Logarithms Common logarithms: log x = log 10 x. Note that

log x = y is equivalent to x = 10 y and the inverse is y = 10 x.

log 2 = 0.30103

log 12.345 = 1.09149

log 0.2001 =

- 0.69875

Graph y = log x. 0 x 10 -5 y 3.

Try any of the above on your calculator if you wish.

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Calculator Evaluation of Logarithms Natural logarithms: ln x = log e x. Note that ln x = y is

equivalent to x = e y and the inverse is y = e x.

ln 2 = 0.69315

ln 12.345 = 2.51325

ln 0.2001 = - 1.60894

Graph y = ln x. 0 x 10 -5 y 2.

Try any of the above on your calculator if you wish.

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Properties Of A Logarithmic Graph.

• Pass through (1, 0). • The graph is continuous.

• The graph is asymptotic to the y axis.

• The graph passes through ( ... , (b –2, -2), (b –1, -1), (b, 1), (b 2, 2), (b 3, 3), …) etc.

• If b > 1 (almost always true) the graph is increasing.

• If 0 < b < 1 the graph is decreasing.

Unless shifted.From the graph y = log x.

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Properties of Logarithmic Functions.

• log b 1 = 0.

• log b b = 1.

• log b b x = x. This is somewhat useful.

•

These four properties follow directly from the definition of logarithm.

blog xb x

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Properties of Logarithmic Functions.

• log b MN = log b M + log b N.

• log b MP = P · log b M.

These four properties are useful in solving logarithmic equations.

• log b M/N = log b M - log b N.

Useful Information

b

log n ln nlog n

log b ln b

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log 7 ln 7log 7 1.209062

log 5 ln 5

This means that1.2090625 7

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Application - Doubling Your MoneyAfter graduating from York College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $5,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 10% a year.

How long will it take for that investment to double?

A = P e rt OR 10,000 = 5000 e 0.10t AND solve for t.

10000/5000 = e 0.10t or 2 = e 0.10t

But 0.10 t = ln 2 so t = ln 2/ .10 = 6.93 years

Remind the students of the “Rule of 72”.ln increase

rt

ln increase

tr

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Application 2Page 121 #103. Using the formula, N = 10 log (I/I 0),

and the fact that I0 = 10 –16 watts/cm2, find the decibel

ratings of the following sounds:

a. Whisper: 10 –13 watts/cm2.

= 10 log 10 3 watts/cm2

= 30

13 2

16 2

10 watts / cmN 10 log

10 watts / cm

N = 10 log (I/I 0)

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Application 2 (continued)Page 121 #103. Using the formula, N = 10 log (I/I 0),

and the fact that I0 = 10 –16 watts/cm2, find the decibel

ratings of the following sounds:

a. Normal conversation: 3.16 · 10 –10 watts/cm2.

= 10 log 3.16 · 10 6 watts/cm2

= 65

NOTE: There are many quantities in real life that are related in a logarithmic manner.

10 2

16 2

3.16 10 watts / cmN 10 log

10 watts / cm

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Ears ringing all over campusHow loud is too loud? Hearing loss occurs after eight hours of exposure

to noises over 85 decibels. The length of exposure is cut in half with each additional three decibels. Any sound above 130 causes immediate harm.

Leaves rustling

10 30 54 63 80 96 120 160

Whisper

Library reading room

Conversation

Traffic at George and Country Club

Car horn

Wiz Khalifa in concert.

Jet engine up close

Carbon 14 Dating

All living things absorb small amounts of radioactive carbon 14 from the atmosphere.

When they die, the carbon 14 stops being absorbed and decays exponentially into ordinary carbon.

Therefore, the proportion of carbon 14 still present in a fossil or other ancient remain can be used to estimate how old it is.

Carbon 14 Dating

The proportion of the original carbon 14 that will be present after t years is

Half-life

Application 3 – DATING BY CARBON 14The Dead Sea Scrolls, discovered in a cave near the Dead Sea in what was then Jordan, are among the earliest documents of Western civilization. Estimate the age of the Dead Sea Scrolls if the animal skins on which some were written contain 78% of their original carbon 14.

Solution:

The proportion of carbon 14 remaining after t years is e–0.00012t and is 78%.

Application 3 – SolutionWe equate this formula to the actual proportion (expressed as a decimal):

e–0.00012t = 0.78

cont’d

ln e–0.00012t = ln 0.78

– 0.00012t = ln 0.78

Therefore, the Dead Sea Scrolls are approximately 2070 years old.

≐ 2071

Take the ln of each side.

Divide by – 0.00012.

And solve..

Simplify.

ln 0.78t

0.00012

ln increase

tr

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Summary.

• We learned about the logarithmic function.

• We learned common logs base 10 and natural logs base e.• We learned about financial problems that involve the logarithmic function.

• We learned about other problems that involve the logarithmic function.

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ASSIGNMENT

§3.4 on my website.7, 8, 9, 10, 11, 12, 13, 14, 15