Logarithms and
Logarithmic Functions
Coach Baughman
November 20, 2003
Algebra II
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Objectives
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
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The students will identify a logarithmic
function. (Knowledge) (Mathematics,
Algebra II, 6.a)
The students will solve logarithmic
expressions. (Application) (Mathematics,
Algebra II, 6.b)
The students will solve logarithmic
functions. (Application) (Mathematics,
Algebra II, 6.c)
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John Napier




Born in Edinburgh, Scotland, in 1550
Began education at St. Andrews
University at the age of 13
Likely acquired mathematical
knowledge at the University of Paris
Died April 4, 1617 in Edinburgh,
Scotland
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Logarithms

Definition: If b and y are positive
where b1, then the logarithm of y
with base b (logby) is defined as
logby = x
if and only if bx = y.
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Special Logarithms


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logb1 = 0
 Why? b0 = 1
logbb = 1
 Why? b1 = b
The logarithm with base 10 is called the
common logarithm. (log10 or log)
The logarithm with base e is called the
natrual logarithm. (loge or ln)
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Examples

Evaluate the expression log381
3x = 81
3x = 34
x = 4

Evaluate the expression log1/28
(1/2)x = 8
(1/2)x
= 23
(1/2)x
= (1/2)-3
x
= -3
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Logarithmic Functions

Exponential functions and
logarithmic functions are inverses


“undo” each other
If g(x) = logbx and f(x) = bx, then
g(f(x)) = logbbx = x and
f(g(x)) = blogbx = x.
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Examples

Simplify the expression 10log2
10log2

=2
Simplify the expression log39x
log39x


= log3(32)x
= log332x
= 2x
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More Examples

Find the inverse of y = log3x
 Use the definition of a logarithm
 y = 3x

Find the inverse of y = ln(x + 1)

y = ln(x + 1)
x = ln(y + 1) (switch x and y)
ex = y + 1 (write in exponential form)

ex – 1 = y
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(solve for y)
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Assessment
1.
2.
3.
4.
5.
6.
Write log7b = 13 in exponential form.
Write 43 = 64 in logarithmic form.
Solve the equation logx(1/32) = -5.
Simplify log5252.
Evaluate log4256.
Find the inverse of y = ln(2x – 5)
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Closing Questions
What did we learn about today?
 Can anyone tell me the definition of
a logarithm?
 Where might you use logarithms?

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