3.2 Logarithms What is a logarithm? · 3.2 Logarithms2015.notebook 5 October 15, 2018 Given...
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3.2 Logarithms2015.notebook 1 October 15, 2018 3.2 Logarithms What is a logarithm?
Transcript of 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
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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
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2.
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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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