8.4 – properties of logarithms · simplifying, expanding, and condensing. ... more properties of...
TRANSCRIPT
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Properties of Logarithms
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Properties of Logarithms
Properties are based off of the rules of exponents (since exponents = logs)
The base of the logarithm can not be equal to 1 and the values must all be positive (no negatives in logs)
𝑏 = 1𝑏 > 0
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Product Rule
logbMN = LogbM + logbN
Ex: logbxy = logbx + logby
Ex: log6 = log 2 + log 3
Ex: log39b = log39 + log3b
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Quotient Rule
Ex:
Ex:
Ex:
yxy
x555 logloglog
P
MN2log
NMN
Mbbb logloglog
5loglog5
log 222 aa
PNM 222 logloglog
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Power Rule
Ex:
Ex:
Ex:
Ex:
BB 5
2
5 log2log
43
7log ba
MxM b
x
b loglog
5log5log 22 xx
ba 77 log4log3
𝑙𝑛 𝑥 =1
2ln(𝑥)
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Pre-Req to Solving Equations: Simplifying, Expanding, and Condensing. Let’s try condensing first…
16log4log 44 nm 22 log4log2 2log5log
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Let’s try some Write the following as a single logarithm.
16log4log 44 2log5log nm 22 log4log2
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Let’s try something more complicated . . .
Condense the logs
log 5 + log x – log 3 + 4log 5
)xlogx(logxloglog 53525 4444
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Let’s try something more complicated . . .
Condense the logs
log 5 + log x – log 3 + 4log 5
= log5𝑥
3+ 𝑙𝑜𝑔54
=log(5𝑥54
3) = log(
𝑥55
3)
𝑙𝑜𝑔4(5
𝑥2) + 𝑙𝑜𝑔4 3𝑥 5 − 𝑙𝑜𝑔4 5𝑥 5
𝑙𝑜𝑔4 5 3𝑥 5 /(𝑥2 5𝑥 5)
)xlogx(logxloglog 53525 4444
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Using Properties to Expand Logarithmic Expressions
Expand:
Use exponential
notation
Use the product rule
Use the power rule
2
1
2 2
1
2 2
log
log
log log
12log log
2
b
b
b b
b b
x y
x y
x y
x y
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Now we’ll try expanding…
Expand
2
4
y3
x10log
3
85
x2log
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Expand
2
4
y3
x10log
3
85
x2log
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More Properties of Logarithms
NMNM aa loglog then , If
NMNM aa then ,loglog If
This one says if you have an equation, you can take
the log of both sides and the equality still holds.
This one says if you have an equation and each side
has a log of the same base, you know the "stuff" you
are taking the logs of are equal.