1. 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

2. 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

3. 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

4. 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

5. 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 log 3
x (log 3 – log 4) = log log 3
log log 3
log 3 – log 4 = x
x  –12.46

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

7. 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

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

9. 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.

10. Homework
study guide