5 minutes Warm-Up Solve each equation for x. Round your answer to the nearest hundredth, if necessary. 1) log x = 3 2) log x = 0.447 3) ln x = 0 4)

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Presentation transcript:

5 minutes Warm-Up Solve each equation for x. Round your answer to the nearest hundredth, if necessary. 1) log x = 3 2) log x = 0.447 3) ln x = 0 4) ln x = 1.61

6.7 Solving Equations and Modeling Objectives: Solve logarithmic and exponential equations by using algebra and graphs Model and solve real-world problems involving logarithmic and exponential relationships

Summary of Exponential-Logarithmic Definitions and Properties Definition of logarithm y = logb x only if by = x Product Property logb mn = logb m + logb n logb = logb m – logb n m n ( ) Quotient Property Power Property logb mp = p logb m

Summary of Exponential-Logarithmic Definitions and Properties Exp-Log Inverse b logb x = x for x > 0 logb bx = x for all x 1-to-1 for Exponents bx = by; x = y 1-to-1 for Logarithms logb x = logb y; x = y logc a = logb a logb c Change-of-Base

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

Example 2 Solve for x. log x + log (x + 3) = 1 log [x(x + 3)] = 1

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

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

Example 4 Suppose 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.

Homework study guide