Properties of Logarithms Section 3.3. Properties of Logarithms What logs can we find using our...

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Properties of LogarithmsSection 3.3
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 changeofbase formula
Properties of LogarithmsChangeofBase 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
Properties of LogarithmsChangeofBase 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
Properties of LogarithmsChangeofBase 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
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 changeofbase formula
4 Log
30 Log 30 Log4 2.4534
Properties of LogarithmsEvaluate the following logarithm using
the natural log function.
14 Log2
14 Log2 0.69315
2.63906
2ln
14ln 3.8073
Properties of LogarithmsEvaluate the following logarithms using
the common log and the natural log.
a)
b)
18 Log5
42 Log2
.79591
3923.5
Properties of LogarithmsWhat is a logarithm?
Therefore, logarithms should have properties that are similar to those of exponents
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
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
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
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
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
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
Properties of LogarithmsSection 3.3
Properties of LogarithmsYesterday:
a) ChangeofBase Formula
b) 3 Properties
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
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
Properties of LogarithmsExpand the following log expression:
y5xlog 34
Start by applying property 1 to separate the product:
y5xlog 34 5log4 3
4xlog ylog4
Properties of LogarithmsExpand the following log expression:
Apply property 3 to eliminate the exponent
y5xlog 34 5log4 3
4xlog ylog4
5log4 xlog3 4 ylog4
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
Properties of LogarithmsExpand the following expression:
Eliminate the exponents
32y4x log 4 log 2 xlog 3y log
4 log xlog2 y log3
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
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
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
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
Properties of LogarithmsExpand the following expression:
2) (x xln 2
212 ] 2) (x [xln
21
2) (x ln x
Properties of Logarithms
21
2) (x ln x
21
2) (x ln ln x
2) (x ln 2
1 ln x
Properties of LogarithmsSection 3.3
Properties of LogarithmsSo far in this section, we have:
a) ChangeofBase Formula
b) 3 Properties
c) Expanded expressions
Today we are going to do the exact opposite
d) Condense expressions
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
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
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
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?
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
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
Properties of Logarithmsln x )2 (x ln 2
Properties of Logarithms
zln 2
1 y log3 x log 2
Properties of Logarithms
2)] (zln y 4ln [2ln x 3
1
Properties of Logarithms
Properties of Logarithms
Properties of Logarithms
Properties of Logarithms
Properties of Logarithms