Unformatted text preview: MAC 1105
Module 9
Exponential and
Logarithmic Functions II
Rev.S08 Learning Objective
Upon completing this module, you should be able to:
1.
2.
3.
4.
5. Learn and apply the basic properties of logarithms.
Use the change of base formula to compute logarithms.
Solve an exponential equation by writing it in logarithmic form
and/or using properties of logarithms.
Solve logarithmic equations.
Apply exponential and logarithmic functions in real world
situations. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 2 Exponential and Logarithmic Functions II
There are two major sections in this module:  Properties of Logarithms
Exponential Functions and Investing Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 3 1 Property 1
• loga(1) = 0 and loga (a) = 1
(1)
• a0 =1 and a 1 = a
• Note that this property is a direct result of the inverse
Note
t he
property loga (ax ) = x
log
• Example: log (1) =0 and ln (e) =1 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 4 Property 2
• loga(m) + loga(n) = loga (mn)
log
• The sum of logs is the log of the product.
The sum
log
• Example: Let a = 2, m = 4 and n = 8
• loga(m) + loga(n) = log2 (4) + log2(8) = 2 + 3
(n) log
• loga(mn) = log2 (4 · 8) = log2(32) = 5
(4 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 5 Property 3
•
• The difference of logs is the log of the quotient.
The difference
log
• Example: Let a = 2, m = 4 and n = 8 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 6 2 Property 4
• • Example: Let a = 2, m = 4 and r = 3
Example: Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 7 Example
• Expand the expression. Write without exponents. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 8 Example
• Write as the logarithm of a single expression
Write
single Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 9 3 Change of Base Formula Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 10 Example of Using the Change of Base
Formula
• Use the change of base formula to evaluate
Use
change
log3 8 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 11 Solve 3(1.2)x + 2 = 15 for x symbolically
Solve 3(1.2)
by Writing it in Logarithmic Form Divide each side by 3
Take common logarithm of each side
(Could use natural logarithm)
Use Property 4: log(m r) = r log (m)
l og
Divide each side by log (1.2)
Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 12 4 Solve ex+2 = 52x for x Symbolically
Solve
Symbolically
by Writing it in Logarithmic Form
Take natural logarithm of each side
Use Property 4: ln (m r) = r ln (m)
ln
ln (e) = 1
S ubtract 2x ln(5) and 2 from each side
Subtract
ln(5)
F actor x from lefthand side
Factor from
Divide each side by 1 – 2 ln (5)
Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 13 Solving a Logarithmic Equation
Symbolically
• In developing countries there is a relationship between
In
t here
the amount of land a person owns and the average
the
daily calories consumed. This relationship is modeled
by the formula C(x ) = 280 ln(x+1) + 1925 where x is the
by
+1)
the
amount of land owned in acres and
S ource:
Source: D. Gregg: The World Food Problem
The • Determine the number of acres owned by someone
whose average intake is 2400 calories per day.
• Must solve for x in the equation
Must
in
280 ln(x+1) + 1925 = 2400
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 14 Solving a Logarithmic Equation
Symbolically (Cont.)
Subtract 1925 from each side Divide each side by 280
E xponentiate each side base e
Exponentiate
I nverse property elnk = k
Inverse
Subtract 1 from each side
Approximate using calculator
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 15 5 Quick Review of
Exponential Growth/Decay Models Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 16 Example of an Exponential Decay:
Carbon14 Dating
The time it takes for half of the atoms to decay into a
h alf
different element is called the halflife of an element
different
halflife
undergoing radioactive decay.
The halflife of carbon14 is 5700 years.
The h alflife
Suppose C grams of carbon14 are present at t = 0.
Suppose
Then after 5700 years there will be C /2 grams present.
Then Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 17 Example of an Exponential Decay:
Carbon14 Dating (Cont.)
Let t be the number of years.
Let A =f ( t ) be the amount of carbon14 present at time t .
Let C be the amount of carbon14 present at t = 0.
Then f(0) = C and f (5700) = C/2.
Thus two points of f are (0,C) and (5700, C /2)
Using the point (5700, C/2) and substituting 5700 for t
and C/2 for A in A = f (t ) = Ca t yields:
C/2 = C a 5700
Dividing both sides by C yields: 1/2 = a 5700
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 18 6 Example of an Exponential Decay:
Carbon14 Dating (Cont.) Halflife Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 19 Radioactive Decay
(An Exponential Decay Model)
If a radioactive sample containing C units has a halflife
h alflife
of k years, then the amount A remaining after x years
of
is given by Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 20 Example of Radioactive Decay
Radioactive strontium90 has a halflife of about 28 years
and sometimes contaminates the soil. After 50 years,
what percentage of a sample of radioactive strontium
would remain?
Note calculator
keystrokes: Since C is present initially and after 50 years .29C remains,
then 29% remains.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 21 7 Example of an Exponential Growth:
Compound Interest
Suppose $10,000 is deposited into an account which pays
5% interest compounded annually. Then the amount A in
5%
c ompounded
Then
the account after t years is:
the
A (t) = 10,000 (1.05)t Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 22 What is the Compound Interest Formula?
• If P dollars is deposited in an account paying an
If
annual rate of interest r , compounded (paid) n times
annual
c ompounded
per year, then after t years the account will contain A
then
dollars, where
•
• annually (1 time per year) • semiannually (2 times per year) • quarterly (4 times per year) • monthly (12 times per year) • Rev.S08 Frequencies of Compounding
(Adding Interest) daily (365 times per year) http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 23 Example: Compounded Periodically
Suppose $1000 is deposited into an account yielding 5%
interest compounded at the following frequencies. How
much money after t years?
much • Annually
• Semiannually
• Quarterly
• Monthly
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 24 8 Example: Compounded Continuously
Suppose $100 is invested in an account with an interest
rate of 8% c ompounded continuously. How much
money will there be in the account after 15 years?
In this case, P = $100, r = 8/100 = 0.08 and t = 15 years.
Thus,
A = P ert
A = $ 100 e.08(15)
A = $ 332.01 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 25 Another Example
• How long does it take money to grow from $100 to
$200 if invested into an account which compounds
$200
c ompounds
quarterly at an annual rate of 5%?
annual
• Must solve for t in the following equation
Must Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 26 Another Example (Cont.) Divide each side by 100
Take common logarithm of each side
Property 4: log(m r) = r log (m)
log
Divide each side by 4log1.0125
Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 27 9 Another Example (Cont.)
Alternatively, we can take natural logarithm of each side
instead of taking the common logarithm of each side.
Divide each side by 100
Take natural logarithm of each side
Property 4: ln (mr) = r ln (m)
ln
Divide each side by 4 ln (1.0125)
Approximate using calculator http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 28 What have we learned?
We have learned to:
1.
2.
3.
4.
5. Learn and apply the basic properties of logarithms.
Use the change of base formula to compute logarithms.
Solve an exponential equation by writing it in logarithmic form
and/or using properties of logarithms.
Solve logarithmic equations.
Apply exponential and logarithmic functions in real world
situations. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 29 Credit
Some of these slides have been adapted/modified in part/whole from the
slides of the following textbook:
• Rockswold, Gary, Precalculus with Modeling and Visualization, 3th
Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 30 10 ...
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Full Document
 Summer '07
 RUSSEL
 Algebra, Logarithmic Functions, Natural logarithm, Logarithm, Gregg, Rockswold, • Quarterly

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