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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 ConsortiumIRIS 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 1 y = log b x is equivalent to b y = x Product PropertyLet b, M, and N be positive real numbers with b 1. log b (MN) = log b M + log b N Quotient PropertyLet b, M, and N be positive real numbers with b 1. Power PropertyLet b and M be positive real numbers with b 1, and let p be any real number. log b M p = p log b M

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Properties of Logarithms (cont.) Exponential-Logarithmic Inverse Properties For b > 0 and b 1, log b b x = x and One-to-One Property of Exponents If b x = b y then x = y One-to-One Property of Logarithms If log b M = log b N, then M = N (M > 0 and N > 0) Change-of-Base FormulaFor any logarithmic bases a and b, and any positive number M,

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log 4 x = 5 4 5 = x x = 1024

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Check your answer! log 4 x = 5 log 4 1024

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ln x = 3 Round your answer to the nearest hundredth. The base of a natural logarithm (ln) is e e 3 = 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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log 5 (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! log 5 (x-6) = -3 log 5 ( - 6) log 5 ( ) 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 e 2 = 3x The base of a natural logarithm (ln) is e x = 2.46

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2log 3 (x-1) = 4 – log 3 5 Write your answer as a decimal rounded to the nearest hundredth. log 3 (x-1) 2 = 4 – log 3 5 log 3 (x-1) 2 + log 3 5 = 4 log 3 5(x-1) 2 = 4

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3 4 = 5(x-1) 2 81 = 5(x 2 – 2x + 1) 81 = 5x 2 – 10x + 5 0 = 5x 2 – 10x - 76 (continued from previous slide) (square (x-1) using FOIL) and 3 4 = 81 Distribute the 5 Subtract 81 from both sides to get the equation equal to zero

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5x 2 – 10x – 76 = 0 Solve using quadratic formula The solution set is x = {5.02, -3.02}

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log 2 (x – 3) + log 2 x – log 2 (x +2) = 2 log 2 x(x – 3) – log 2 (x +2) = 2

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(continued from previous slide) 2 2 = 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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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 x 3 = log 64 x 3 = 64 x = 4

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

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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! 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 – 12 = 108 4x = 120 x = 30 4x = 120 x = 30

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

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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 2.82 x 10 22 ergs.

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