Presentation on theme: "Solving Logarithmic Equations"— Presentation transcript:
1 Solving Logarithmic Equations Math 1111Tosha Lamar, Georgia Perimeter College Online
2 Who would ever need to solve a logarithmic equation? There are several areas in which logarithmic equations must be used such asAn 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
3 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
4 1995 Tokyo EarthquakeThe 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!
5 Properties of Logarithms From our previous lessons, recall the properties of logarithms.We will refer back to these throughout the lesson.Properties of LogarithmsDefinition of LogarithmFor x > 0 and b > 0, b 1y = logb x is equivalent to by = xProduct PropertyLet b, M, and N be positive real numbers with b 1.logb (MN) = logb M + logb NQuotient PropertyPower PropertyLet b and M be positive real numbers with b 1, and let p be any real number.logb Mp = p logb M
6 logb bx = x and Properties of Logarithms (cont.) Exponential-Logarithmic Inverse PropertiesFor b > 0 and b 1,logb bx = x andOne-to-One Property of ExponentsIf bx = by then x = yOne-to-One Property of LogarithmsIf logb M = logb N, then M = N(M > 0 and N > 0)Change-of-Base FormulaFor any logarithmic bases a and b, and any positive number M,
9 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 ee3 = xx = 20.09(rounded to the nearest hundredth)
10 Check your answer! ln x = 3 ln 20.09 It is not “exactly” 3 since you rounded the answer to the nearest hundedth.
11 log5 (x-6) = -3Write your answer as a fraction in lowest terms.5-3 = x – 6= x – 6
27 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.
28 (because log 10 = 1)Multiply by (3/2) to get rid of the fraction on the right(3/2) * 7.1 = 10.65Add 11.8 to both sides
29 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.