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**Solving Logarithmic Equations**

Math 1111 Tosha 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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1995 Tokyo Earthquake 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!

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**Properties of Logarithms**

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 1 y = 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 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

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**logb bx = x and 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,

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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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**e3 = x x = 20.09 ln x = 3 The “base” of a natural logarithm (ln) is e**

Round 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 ( ) log5 ( ) Correct!!

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5 ln (3x) = 10 Write your answer as a decimal rounded to the nearest hundredth. Divide both sides of the equation by 5 ln (3x) = 2 The “base” of a natural logarithm (ln) is e e2 = 3x x = 2.46

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**2log3 (x-1) = 4 – log3 5 log3 (x-1)2 = 4 – log3 5**

Write 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**

(continued from previous slide) 34 = 5(x-1)2 81 = 5(x2 – 2x + 1) (square (x-1) using FOIL) and 34 = 81 81 = 5x2 – 10x + 5 Distribute the 5 0 = 5x2 – 10x - 76 Subtract 81 from both sides to get the equation equal to zero

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**The solution set is x = {5.02, -3.02}**

5x2 – 10x – 76 = 0 Solve using quadratic formula 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

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**(continued from previous slide)**

22 = 4 Multiply both sides by (x+2) Distribute the 4 Subtract 4x and subtract 8 to get one side of the equation equal to zero

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**Set each factor equal to zero (zero product property)**

x = 8 x = -1 Solution set: x = {8, -1}

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3 log x = log 64 log x3 = log 64 x3 = 64 x = 4

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2 log x – log 6 = log 96 log x2 – log 6 = log 96

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Multiply both sides by 6 Why 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!

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**log(x – 3) + log 4 = log 108 log 4(x – 3) = log 108**

4x = x = 30

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**log2(x + 12) – log2 (x + 3) = log2 (x - 4)**

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**(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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**log2(6 + 12) – log2 (6 + 3) = log2 (6 - 4)**

Always check your answers Check x = 6 log2(6 + 12) – log2 (6 + 3) = log2 (6 - 4) log2(18) – log2 (9) = log2 (2) log2(2)= log2 (2) Check x = -4 log2( ) – log2 (-4 + 3) = log2 (-4 - 4) log2(8) – log2 (-1) = log2 (-8) CANNOT TAKE LOG OF NEGATIVE NUMBER ! FINAL ANSWER: x = 6

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**Now we can go back to our earthquake problem**

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.

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(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

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**Therefore, the amount of energy, E , released by this earthquake was 2**

Therefore, the amount of energy, E , released by this earthquake was 2.82 x ergs.

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