properties of logarithms section 3.3. properties of logarithms what logs can we find using our...
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![Page 1: Properties of Logarithms Section 3.3. Properties of Logarithms What logs can we find using our calculators? ◦ Common logarithm ◦ Natural logarithm Although](https://reader030.vdocument.in/reader030/viewer/2022032611/56649efa5503460f94c0c8d1/html5/thumbnails/1.jpg)
Properties of LogarithmsSection 3.3
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Properties of LogarithmsWhat logs can we find using our calculators?
◦Common logarithm◦Natural logarithm
Although these are the two most frequently used logarithms, you may need to evaluate other logs at times
For these instances, we have a change-of-base formula
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Properties of LogarithmsChange-of-Base Formula
Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows:
x Loga
aa Log
xLog x Log
b
b
Base b
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Properties of LogarithmsChange-of-Base Formula
Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows:
x Loga
aa Log
xLog x Log
10
10
Base 10
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Properties of LogarithmsChange-of-Base Formula
Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows:
x Loga
aa ln
ln x x Log
Base e
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Properties of LogarithmsEvaluate the following logarithm:
30 Log4
→ 4 raised to what power equals 30?
Since we don’t know the answer to this, we would want to use the change-of-base formula
4 Log
30 Log 30 Log4 2.4534
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Properties of LogarithmsEvaluate the following logarithm using
the natural log function.
14 Log2
14 Log2 0.69315
2.63906
2ln
14ln 3.8073
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Properties of LogarithmsEvaluate the following logarithms using
the common log and the natural log.
a)
b)
18 Log5
42 Log2
.79591
3923.5
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Properties of LogarithmsWhat is a logarithm?
Therefore, logarithms should have properties that are similar to those of exponents
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Properties of LogarithmsFor example, evaluate the following:
a)
b)
c)
52 x x 5 2x 7x
37 x x 3 7x 4x
27 )(x 2 7x 14x
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Properties of LogarithmsJust like we have properties for
exponents, we have properties for logarithms.
These properties are true for logs with base a, the common logs, and the natural logs
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Properties of LogarithmsProperties of LogarithmsLet a be a positive number such that a ≠ 1, and
let n be a real number. If u and v are positive real numbers, the following properties are true.
(uv)Log 1) a vLog u Log aa
v
uLog 2) a vLog u Log aa
nu Log 3) a u Logn a
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Properties of LogarithmsUse the properties to rewrite the
following logarithm:
10zLog3
From property 1, we can rewrite this as the following:
10zLog3 10Log3 z Log3
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Properties of LogarithmsUse the properties to rewrite the
following logarithm:
2
y Log10
From property 2, we can rewrite this as the following:
2
yLog10 y Log10 2 Log10
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Properties of LogarithmsUse the properties to rewrite the
following logarithm:
36 z
1 Log
From property 3, we can rewrite this as the following:
36 z
1Log z Log -3
6 z Log3 6
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Properties of LogarithmsSection 3.3
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Properties of LogarithmsYesterday:
a) Change-of-Base Formula
b) 3 Properties
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Properties of LogarithmsToday we are going to continue working
with the three properties covered yesterday.
(uv)Log 1) a vLog u Log aa
v
uLog 2) a vLog u Log aa
nu Log 3) a u Logn a
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Properties of LogarithmsThese properties can be used to rewrite
log expressions in simpler terms
We can take complicated products, quotients, and exponentials and convert them to sums, differences, and products
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Properties of LogarithmsExpand the following log expression:
y5xlog 34
Start by applying property 1 to separate the product:
y5xlog 34 5log4 3
4xlog ylog4
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Properties of LogarithmsExpand the following log expression:
Apply property 3 to eliminate the exponent
y5xlog 34 5log4 3
4xlog ylog4
5log4 xlog3 4 ylog4
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Properties of LogarithmsExpand the following expression:
32y4x log
Start by applying property 1 to separate the product:
32y4x log 4 log 2 xlog 3y log
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Properties of LogarithmsExpand the following expression:
Eliminate the exponents
32y4x log 4 log 2 xlog 3y log
4 log xlog2 y log3
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Properties of LogarithmsRewrite the following logarithm:
7
5 -3x ln
For problems involving square roots, begin by converting the square root to a power
7
5 -3x ln
7
5 x 3ln
2
1
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Properties of Logarithms
7
5 x 3ln
2
1
Apply property 1 to get rid of the quotient:
7
5 x 3ln
2
1
2
1
5) -(3x ln 7ln
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Properties of Logarithms
2
1
5) -(3x ln 7ln
Apply property 3 to get rid of the exponent
2
1
5) -(3x ln 7ln 5) -(3x ln 2
1 7ln
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Properties of LogarithmsRewrite the following logarithmic
expressions:
2y
5 x ln
zy2xln 23 zln y 2ln 3ln x 2ln
y2ln 5) (x ln 2
1
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Properties of LogarithmsExpand the following expression:
2) (x xln 2
212 ] 2) (x [xln
21
2) (x ln x
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Properties of Logarithms
21
2) (x ln x
21
2) (x ln ln x
2) (x ln 2
1 ln x
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Properties of LogarithmsSection 3.3
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Properties of LogarithmsSo far in this section, we have:
a) Change-of-Base Formula
b) 3 Properties
c) Expanded expressions
Today we are going to do the exact opposite
d) Condense expressions
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Properties of LogarithmsWhen we were expanding, what order did
we typically apply the properties in?◦ Property 1 or Property 2◦End with Property 3
When we condense, we use the opposite order◦Property 3◦Property 1 or Property 2
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Properties of LogarithmsThe most common error:
Log x – Log y
When you condense, you are condensing
the expression down to one log function
y Log
xLog
y
x Log
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Properties of LogarithmsCondense the following expression:
1) (x log 3 x log 2
1
Start by applying property 3, then move on to properties 1 and 2
32
1
1) (x log xlog
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Properties of Logarithms
32
1
1) (x log xlog
31) (x log x log
31) (x x log
Is this expression simplified to one log function?
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Properties of LogarithmsCondense the following expression:
1)] (x log x [log 3
122
3
1
22 1)] (x log x log[
3
1
2 1)] x(x log[
32 1) x(x log
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Properties of Logarithms
ln x )2 (x ln 2
2)] (zln y 4ln [2ln x 3
1
zln 2
1 y log3 x log 2
x
2) (x ln
2
y
zx log
3
2
2) (zy
xln 3
4
2
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Properties of Logarithmsln x )2 (x ln 2
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Properties of Logarithms
zln 2
1 y log3 x log 2
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Properties of Logarithms
2)] (zln y 4ln [2ln x 3
1
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Properties of Logarithms
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Properties of Logarithms
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Properties of Logarithms
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Properties of Logarithms
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Properties of Logarithms