3.2 logarithms what is a logarithm? · 3.2 logarithms2015.notebook 5 october 15, 2018 given...
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3.2 Logarithms
What is a logarithm?
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A logarithm is an EXPONENT
y = log a x
log a x = y equivalent
it is said, "log base a of x is y."
Logarithmic form
Exponential form a y = x
∴ log a x = y and a y = x
ARE EQUIVALENT
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Given logarithmic form change to exponential form
1.
3.
2.
32=9
e0=1
23=1/8
4. 102=100
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Given exponential form change to logarithmic formGo the other way
1. 2.
log4 64=3 log100 10 = 1/2
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1. Describe the transformation: f(x) = 4 x is the the mother function.a. f(x) = 4x 3b. f(x) = 4 (x2)
c. f(x) = 4 (x+3) 4
2. Write as a logarithmic equation.a. 42=16b. 63= 216c 84=4096
3. Write as an exponential equation.a. log3 81=4b. log5 125 = 3c. log41024=5
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Solve for x
We can generalize for all logarithms...
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Write as an exponent: log381 = 4
Write as a logarithm: 82 = 64
Evaluate:log5125log232log31log42log .01log70log2-4
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Properties of Logarithms
1.
2.
3.
4.
These properties are going to help us solve equations. We will be able to replace terms and parts of terms...be patient you will see in 3.5.
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Logarithmic Functions and Their GraphsGraphs of logarithms:
Graph: Change of base!!
Domain=_________Range=__________xintercept=_______yintercept=_______
x=2y
Without a calculator, rewrite as an exponential, then substitute.Now graph y = 2 x
What can you tell me about the two graphs?
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Inversescheck it out when both in exponential form.
Reflect the y = x line. (axis of symmetry!)
Solving: f(x) = log3 x What is its inverse?
Switch the x and y and solve for y!
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Domain=_________Range=__________xintercept=_______yintercept=_______
f(x) = log3x
What is the inverse?
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Domain=_________Range=__________xintercept=_______yintercept=_______
What is the inverse?
No calculator
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Domain=_________Range=__________xintercept=_______yintercept=_______
What is the inverse?
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The graph has the yaxis as a vertical asymptote.
Does that make sense?
If you put the equation in exponential form, what can you put for y to make x negative or zero?
NOTHING!!
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The Natural Logarithmic Function (works the exact same way)
Just remember...
it's easier to manipulate when you see the base
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1.
2.
3.
4.
and
Properties of Natural Logarithms
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Calculator only evaluates logs in two bases.
Base 10 and Base e
Evaluate and round to three decimal places
ln10
Must use the change of base formula for all others!!! log5125= ?
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The relationship between the number of decibels β (named after Alexander Graham BELL ) and the intensity of sound I in watts/m2 is given by:β = 10 log (I/10-12)a. Determine the number of decibels of a sound with an intensity of 1 watt/m2
b. Determine the number of decibels of a sound with an intensity of 10-2 watt/m2
c. The intensity of sound in part a is 100 times as great as that in part b. Is the number of decibels 100 times as great? Explain.
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Transformations of Graphs of Logs
(same rules apply for shifting and reflecting) State the domain and range for the following functions.
f(x) = log3 (x+2) + 5
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Sketch a graph of: y = - logx + 4 (by hand)
Asymptote:Intercept:Domain:Range:
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