4.4 Evaluate Logarithms and Graph Logarithmic Functions

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4.4 Evaluate Logarithms and Graph Logarithmic
Functions

• Part 2

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Definition

• Logarithms are the "opposite" of exponentials,
• Logs "undo" exponentials.
• Logs are the inverses of exponentials.

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Writing Logarithms
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You read it Log base b of a equals c
log is the operation b is the base a is
the object of the log c is what you get when
you evaluate the log
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Exponential Form
x
y

b
Logarithmic Form
log
x
y

b
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Evaluating logarithms now you try some!

• Log 4 16
• Log 5 1
• Log 16 4
• Log 3 (-1)
• (Think of the graph of y 3x)

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0
½ (because 161/2 4)
undefined
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You should learn the following general forms!!!

• Log a 1 0 because a0 1
• Log a a 1 because a1 a
• Log a ax x because ax ax

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Common logarithms

• log x log 10 x
• Understood base 10 if nothing is there.

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Common Logs and Natural Logs with a calculator
log10 button lne button
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Finding Inverses

• Find the inverse of
• y log3x
• By definition of logarithm, the inverse is y3x
• OR write it in exponential form and switch the x
  y!
• 3y x 3x y

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Example 1

• Write 53 125 in logarithmic form.
• Write log381 4 in exponential form.

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Try This Complete the table.
1
2
3
4
Exponential Form 25 32 3-2 1/9
Logarithmic Form log101000 3 Log164 1/2
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Lets look at their graphs
y x
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To Evaluate Logs without a Calculator

• Change the log to an exponential.

1. log2 32 x
2. log4 2 x
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Solve for x.
Change the log to an exponential.
1. log2 64 x
2. logx 343 3
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Evaluate without a calculator
Change the log to an exponential.

• log 2 8 x
• log 2 1 x
• 3. Find the value of k k log 9 3
• 4. Find the value of k ½ log k 9
• 5. Find the value of k 3 log 7 k

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Common Logarithms
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• Logarithms with base ______ are called common
  logarithms.
• Sometimes the base is assumed and not written.
• Thus, if you see a log written without a base,
  you assume the base is _______.
• The log button the calculator uses base _____.

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10
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Use your calculator to evaluate
1.71

• log 51
• log 4
• log 0.215

Which means
0.6
0.67
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Do You Know What X is?
Change the exponential to a log. Then use
calculator.
4. Solve for x 10x 728 5. Solve
for x
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Remember e ?
The Natural Base
Used for applications with CONTINUOUS growth or
decay!
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Natural Logarithm

• A natural logarithm is a logarithm with base e,
  denoted by ln.
• A natural logarithm is the inverse of an
  exponential function with base e.

Exponential Form
Logarithmic Form
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Lets look at their graphs
y x
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Evaluate f(x)ln x to the nearest thousandth for
each value of x below
? (see graph)
0.693
0.693
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• 13. Find the inverse of y ln(x1)
• 14. Find the inverse of y 5x .

y ex - 1
y log5x
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Homework

• Book
• Pg. 147 16 - 24 all
• Pg. 148 13 21 all