# Properties of Logarithmic Functions Properties of Logarithmic Functions Objectives: Simplify and...

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Properties of Logarithmic FunctionsObjectives: Simplify and evaluate expressions involving logarithmsSolve equations involving logarithms

Properties of LogarithmsProduct Propertylogb (mn) = logb m + logb nFor m > 0, n > 0, b > 0, and b 1:

Example 1= log5 12 + log5 10log5 120 = given: log5 12 1.5440log5 10 1.4307 log5 (12)(10) 1.5440 + 1.4307 2.9747

Properties of LogarithmsQuotient PropertyFor m > 0, n > 0, b > 0, and b 1:

Example 2given: log5 12 1.5440log5 10 1.4307log5 1.2 = log5 12 log5 10 1.5440 1.4307 0.1133

Properties of LogarithmsPower PropertyFor m > 0, n > 0, b > 0, and any real number p:logb mp = p logb m

Example 3given: log5 12 1.5440log5 10 1.4307log5 1254 = 4 3= 12= 4 log5 1255x = 12553 = 125x = 3

PracticeWrite each expression as a single logarithm.1) log2 14 log2 72) log3 x + log3 4 log3 23) 7 log3 y 4 log3 x

Warm-UpWrite each expression as a single logarithm. Then simplify, if possible.4 minutes1) log6 6 + log6 30 log6 52) log6 5x + 3(log6 x log6 y)

Properties of LogarithmsExponential-Logarithmic Inverse Propertylogb bx = xFor b > 0 and b 1:and b logbx = x for x > 0

Example 1Evaluate each expression.a)b)

PracticeEvaluate each expression.1) 7log711 log3 812) log8 85 + 3log38

Properties of LogarithmsOne-to-One Property of LogarithmsIf logb x = logb y, then x = y For b > 0 and b 1:

Example 2Solve log2(2x2 + 8x 11) = log2(2x + 9) for x. log2(2x2 + 8x 11) = log2(2x + 9)2x2 + 8x 11 = 2x + 92x2 + 6x 20 = 02(x2 + 3x 10) = 02(x 2)(x + 5) = 0x = -5,2Check:log2(2x2 + 8x 11) = log2(2x + 9)log2 (1) = log2 (-1)undefinedlog2 13 = log2 13true

PracticeSolve for x.1) log5 (3x2 1) = log5 2x2) logb (x2 2) + 2 logb 6 = logb 6x

Solving Equations and ModelingObjectives: Solve logarithmic and exponential equations by using algebra and graphsModel and solve real-world problems involving logarithmic and exponential relationships

Summary of Exponential-Logarithmic Definitions and PropertiesDefinition of logarithmy = logb x only if by = x Product Propertylogb mn = logb m + logb n Quotient PropertyPower Propertylogb mp = p logb m

Summary of Exponential-Logarithmic Definitions and PropertiesExp-Log Inverseb logb x = x for x > 0 logb bx = x for all x 1-to-1 for Exponentsbx = by; x = y 1-to-1 for Logarithmslogb x = logb y; x = yChange-of-Base

Example 1Solve for x.3x 2 = 4x + 1log 3x 2 = log 4x + 1(x 2) log 3 = (x + 1) log 4x log 3 2 log 3 = x log 4 + log 4x log 3 x log 4 = log 4 + 2 log 3x (log 3 log 4) = log 4 + 2 log 3x 12.46

Example 2Solve for x.log x + log (x + 3) = 1log [x(x + 3)] = 1101 = x(x + 3)x2 + 3x 10 = 0(x + 5)(x 2) = 0x = 2,-5101 = x2 + 3x

Example 2Solve for x.log x + log (x + 3) = 1log x + log (x + 3) = 1x = 2,-5Let x = 2log 2 + log (2 + 3) = 1log 2 + log 5 = 11 = 1log x + log (x + 3) = 1Let x = -5log -5 + log (-5 + 3) = 1log -5 + log -2 = 1undefinedx = 2Check:

Example 3Solve for x.8e2x-5 = 56e2x-5 = 7ln e2x-5 = ln 72x - 5 = ln 7x 3.47

Example 4Suppose that the magnitude, M, of an earthquake measures 7.5 on the Richter scale. Use the formula below to find the amount of energy, E, released by this earthquake.