Transcript Exponential and Logarithmic Function Lesson 2.4

Exponential and Logarithmic
Function
Lesson 2.4
Animated Exponential Function
A^x
8
Go to Spreadsheet of ax for
a between 1/2 and 2
-5
-3
-1
1
-2
3
5
Exponential Function
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Definition:
Where:
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f ( x)  a  b x
b > 0 is a constant
b≠1
a is a constant
b<1
a
b>1
a
Reminder
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Properties of exponents
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Note Theorem 2.8, pg 81
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Properties of exponential functions
Example
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Consider equation:
Solution:
4
x2  x
 16
x2  x
2
4
4
Thus
x x2
2
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Note that calculator
will solve also
Logarithmic Functions
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Inverse of exponential functions
log a x  y
 ay  x
f ( x)  y  f 1 ( y)  x
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Learn to translate back and forth from one
form to the other
log 3 9  2  32  9
Examples
1
log 2  ?
8
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Try these:
log 3 27  ?
10 4 x 5  47
solve
Solutions
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Check your work
1
log 2  3
8
log 3 27  3
104 x 5  47
log10 47  4 x  5
Natural Log
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Often the number e  2.71828… is used as
the base for logarithms
logex = ln x
This is the inverse of the exponential
function with e as the base
ye
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x
 ln y  x
Many computer languages use exp(x) for ex
Changing Bases
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To change from base b to base a, use the
formula shown:
log a x
log b x 
log a b
log 2 7  ?
Hint: base a will
be base 10
Using Logarithms
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Use to solve exponential equations
5x3  40
log(5x 3 )  log 40
( x  3)  log 5  log 40
log 40
x3
log 5
Application of Exponents
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Exponential Growth
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y = P0 ax
periodic exponential growth
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y = P0 ekt
continuous exponential growth
Application of Exponents
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Annually compounded interest
A = (1 + r)t
Compounded n times per year
 r
A  P 1  
 n
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nt
Continuous compounding
A = P ert
Assignment
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Lesson 2.4
Page 89
Exercises 1 – 39 odd, 55 – 69 odd