Exponential and Logarithmic Functions

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ExponentialandLogarithmic Functions
By Jeffrey Bivin Lake Zurich High
School jeff.bivin_at_lz95.org
Last Updated January 30, 2008
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With your Graphing Calculatorgraph each of the
following
y 2x
y 3x
y 5x
y 1x
Determine what is happening when the base is
changing in each of these graphs.
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y 2x
y 3x
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y 2x
y 3x
y 5x
y 1x
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y 2x
y 3x
y 5x
y 4x
y 10x
y (3/2)x
Determine where each of the following would
lie? y10x y4x y (3/2)x
y 1x
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Exponential graphs with translations
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f(x) 2x
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f(x) 2x-3
x - 3 0
x 3
(3, 1)
3
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f(x) 2x2 - 3
x 2 0
x -2
3
2
(-2, -2)
y -3
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f(x) -(2)x-4 2
x - 4 0
x 4
4
2
y -2
(4, -3)
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Compound Interest
You deposit 5000 into an account that pays 4.5
interest. What is the balance of the account
after 10 years if the interest is compounded
quarterly?
A Final amount unknown
P Principal 5000
r rate of interest .045
n number of times compounded per year 4
t number of years compounded 10
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Compound Interest
You deposit 5000 into an account that pays 4.5
interest. What is the balance of the account
after 10 years if the interest is compounded
quarterly?
A unknown
P 5000
r .045
n 4
t 10
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Compound Interest
You deposit 5000 into an account that pays 4.5
interest. What is the balance of the account
after 10 years if the interest is compounded
quarterly?
weekly?
A unknown
P 5000
r .045
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n 4
t 10
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Exponential
DECAY
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With your Graphing Calculatorgraph each of the
following
y (1/2)x
y (1/3)x
y 1x
Determine what is happening when the base is
changing in each of these graphs.
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y 2x
y 5x
y (1/3)x
y 3x
y (½)x
y 1x
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f(x) 2-x (1/2)x
(0, 1)
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f(x) (½)x-3 - 2 (2)-x3 - 2
x - 3 0
x 3
3
2
(3, -1)
y -2
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A new Number
We could use a spreadsheet to determine an
approximation.
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A new Number
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y 2x
y 3x
Graph y ex
y ex
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Graph y ex2
y ex2
y ex
x 2 0
x -2
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Compound Interest-continuously
You deposit 5000 into an account that pays 4.5
interest. What is the balance of the account
after 10 years if the interest is compounded
continuously?
A Final amount unknown
P Principal 5000
r rate of interest .045
t number of years compounded 10
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Compound Interest-continuously
You deposit 5000 into an account that pays 4.5
interest. What is the balance of the account
after 10 years if the interest is compounded
continuously?
A unknown
P 5000
r .045
t 10
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Bacteria Growth
You have 150 bacteria in a dish. It the constant
of growth is 1.567 when t is measured in hours.
How many bacteria will you have in 7 hours?
y Final amount unknown
n initial amount 150
k constant of growth 1.567
t time 7
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Bacteria Growth
You have 150 bacteria in a dish. It the constant
of growth is 1.567 when t is measured in hours.
How many bacteria will you have in 7 hours?
y unknown
n 150
k 1.567
t 7
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y 2x
x 2y
INVERSE
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y 2x
x 2y
INVERSE
How do we solve this exponential equation for the
variable y?
?
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LOGARITHMS
exponential
logarithmic
b gt 0
A gt 0
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exponential
logarithmic
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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y 2x
x 2y
INVERSE
ylog2x
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x 2y
y log2x
y log3x
y log5x
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x (½)y
y log½x
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Solve for x
log2(x5) 4
24 x 5
16 x 5
11 x
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Solve for x
logx(32) 5
x5 32
x5 25
x 2
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Evaluate
log3(25) u
3u 25
3u 52
??????
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Change of Base Formula
if b 10
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Evaluate
log3(25)
2.930
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Evaluate
log5(568)
3.941
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Properties of Logarithms

• Product Property
• Quotient Property
• Power Property
• Property of Equality

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Product Property
multiplication
addition
multiplication
addition
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Product Property
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Quotient Property
division
subtraction
division
subtraction
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Quotient Property
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Power Property
p
logb(m p )
logb(mp ) plogb(m)
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Power Property
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Property of Equality
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Expand
product property
power property
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Expand
quotient property
product property
power property
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Expand
quotient property
product property
distributive property
power property
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Condense
(
(
power property
product property
quotient property
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Condense
(
(
Power property
group / factor
product property
quotient property
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Condense
Power property
(
(
re-organize
group
product property
quotient property
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Solve for x
Property of Equality
We must check for false solutions!
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Solve for x
checks!
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Solve for x
Solution
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Solve for n
Condense left side
Property of Equality
We must check for false solutions!
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Solve for n
checks!
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Solve for n
Solution
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Solve for x
Condense left side
Convert to exponential form
We must check for false solutions!
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Solve for x
fails
The argument must be positive
checks!
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Solve for x
Solution
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Solve for x
Convert to exponential form
We must check for false solutions!
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Solve for x
checks!
checks!
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Solve for x
Solution
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Solve for x
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Solve for x
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Solve for x
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Solve for x
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Solve for x
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Solve for x
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Solve for x
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Solve for x
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