Solving Logarithmic Equations Math 1111 Tosha Lamar, Georgia Perimeter College Online.
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![Page 1: Solving Logarithmic Equations Math 1111 Tosha Lamar, Georgia Perimeter College Online.](https://reader038.fdocuments.in/reader038/viewer/2022102705/551a4365550346cb358b5719/html5/thumbnails/1.jpg)
Solving Logarithmic Equations
Math 1111Tosha Lamar, Georgia Perimeter College Online
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Who would ever need to solve a logarithmic equation?
There are several areas in which logarithmic equations must be used such as
An archeologist wants to know how old a fossil is using the measure of the Carbon-14 remaining in a dinosaur bone.
A meteorologist needs to measure the atmospheric pressure at a given altitude.
A scientist needs to know how long it will take for a certain liquid to cool from 75 degrees to 50 degrees.
We need to measure how loud (in decibels) a rock concert is to determine if it could cause long-term hearing loss.
. . . just to name a few
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When we finish this lesson we will be able to solve a problem like this:
In 1989, an earthquake that measures 7.1 on the Richter scale occurred in San Fransico, CA. Find the amount of energy, E , released by this earthquake.
On the Richter scale, the magnitude, M, of an earthquake depends on the amount of energy, E, released by the earthquake according to this formula. M is a number between 1 and 9. A destructive earthquake usually measures greater than 6 on the Richter scale.
Illustration from IRIS Consortium
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The amount of energy, measured in ergs, is based on the amount of ground motion recorded by a seismograph at a known distance from the epicenter of the earthquake.
After we have practiced solving some logarithmic equations we will go back and find the answer to this problem!
1995 Tokyo Earthquake
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From our previous lessons, recall the properties of logarithms.We will refer back to these throughout the lesson.
Properties of Logarithms
Definition of Logarithm For x > 0 and b > 0, b 1y = logb x is equivalent to by = x
Product Property Let b, M, and N be positive real numbers with b 1.logb (MN) = logb M + logb N
Quotient Property Let b, M, and N be positive real numbers with b 1.
Power Property Let b and M be positive real numbers with b 1, and let p be any real number.logb Mp = p logb M
NMN
Mbbb logloglog
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Properties of Logarithms (cont.)
Exponential-Logarithmic Inverse Properties
For b > 0 and b 1, logb bx = x and
One-to-One Property of Exponents
If bx = by then x = y
One-to-One Property of Logarithms
If logb M = logb N, then M = N(M > 0 and N > 0)
Change-of-Base Formula For any logarithmic bases a and b, and any positive number M,
b
MM
a
ab log
loglog
xb xb log
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log4 x = 5
45 = x
x = 1024
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Check your answer!
log4 x = 5
log4 1024
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ln x = 3Round your answer to the nearest hundredth.
The “base” of a natural logarithm (ln) is e
e3 = x
x = 20.09(rounded to the nearest hundredth)
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Check your answer!
ln x = 3
ln 20.09 It is not “exactly” 3 since you rounded the answer to the nearest hundedth.
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log5 (x-6) = -3 Write your answer as a fraction in lowest terms.
5-3 = x – 6
= x – 6
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Check your answer!
log5 (x-6) = -3
log5 ( - 6) 125
751
log5 ( ) 125
1
Correct!!
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5 ln (3x) = 10Write your answer as a decimal rounded to the nearest hundredth.
Divide both sides of the equation by 5
ln (3x) = 2
e2 = 3x
The “base” of a natural logarithm (ln) is e
x = 2.46
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2log3 (x-1) = 4 – log3 5Write your answer as a decimal rounded to the nearest hundredth.
log3 (x-1)2 = 4 – log3 5
log3 (x-1)2 + log3 5 = 4
log3 5(x-1)2 = 4
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log3 5(x-1)2 = 4
34 = 5(x-1)2
81 = 5(x2 – 2x + 1)
81 = 5x2 – 10x + 5
0 = 5x2 – 10x - 76
(continued from previous slide)
(square (x-1) using FOIL) and 34 = 81
Distribute the 5
Subtract 81 from both sides to get the equation equal to zero
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a
acbbx
2
42
5x2 – 10x – 76 = 0 Solve using quadratic
formula
)5(2
)76)(5(4)10()10( 2 x
10
162010x
The solution set is x = {5.02, -3.02}
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log2 (x – 3) + log2x – log2 (x +2) = 2
log2 x(x – 3) – log2 (x +2) = 2
2)2(
)3(log2
x
xx
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2)2(
)3(log2
x
xx(continued from previous slide)
22 = 4
Multiply both sides by (x+2)
)2(
)3(22
x
xx
)2(
)3(4
x
xx
)3()2(4 xxx
xxx 384 2
870 2 xx
Distribute the 4
Subtract 4x and subtract 8 to get one side of the equation equal to zero
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870 2 xx
)1)(8(0 xx
0108 xx
x = 8
x = -1
Solution set: x = {8, -1}
Set each factor equal to zero (zero product property)
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3 log x = log 64
log x3 = log 64
x3 = 64
33 3 64xx = 4
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2 log x – log 6 = log 96
log x2 – log 6 = log 96
96log6
log2
x
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96log6
log2
x
966
2
x
5762 x
24xWhy not also -24??
Look at the original equation – if we allowed x to be -24 we would be taking the log of a negative number!
Multiply both sides by 6
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log(x – 3) + log 4 = log 108
log 4(x – 3) = log 108
log (4x – 12) = log 108
4x – 12 = 108
4x = 120 x = 30
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log2(x + 12) – log2 (x + 3) = log2 (x - 4)
)4(log3
12log 22
xx
x
43
12
xx
x
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)3)(4(3
123
xxx
xx
1212 2 xxx
2420 2 xx
)4)(6(0 xx
606 xx
404 xx
(multiply both sides by (x+3)
Multiply (x-4)(x+3) using FOIL
Subtract x and subtract 12 from both sides to get the equation equal to zero
Factor
Set each factor equal to zero
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Always check your answers
log2(6 + 12) – log2 (6 + 3) = log2 (6 - 4)
log2(18) – log2 (9) = log2 (2)
Check x = -4
CANNOT TAKE LOG OF NEGATIVE NUMBER !
log2(-4 + 12) – log2 (-4 + 3) = log2 (-4 - 4)
log2(8) – log2 (-1) = log2 (-8)
Check x = 6
log2(2)= log2 (2)
FINAL ANSWER: x = 6
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Now we can go back to our earthquake problem!
In 1989, an earthquake that measures 7.1 on the Richter scale occurred in San Fransico, CA. Find the amount of energy, E , released by this earthquake.
)10log(log3
21.7 8.11 E
)10log8.11(log3
21.7 E
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)8.11(log3
21.7 E
)8.11(log3
2
2
31.7
2
3 E
8.11log65.10 E
Elog45.22
(because log 10 = 1)
Multiply by (3/2) to get rid of the fraction on the right
(3/2) * 7.1 = 10.65
Add 11.8 to both sides
E45.2210
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Therefore, the amount of energy, E , released by this earthquake was 2.82 x 1022
ergs.