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8.6 Natural Logarithms Learning Objective: After this lesson, I will be able to solve equations using natural logs. Language Objective: After this lesson, I will be able to explain how to switch between exponential functions with base e and natural logs. What is “e” ? e ≈ 2.71828 Try these on your calculator: 1. e5 2. 2e3 Natural logarithmic function: 3. e-1 4. 1 e loge x = y is commonly written as ln x = y Y = ex and y = ln x are inverses. Simplifying Natural Logs: You can use the same product, quotient, and power rules with natural logs as with other logs. Examples: Write as a single log. 1. 3 ln 6 – ln 8 2. 3 ln x + ln y Examples: Expand. x 1. ln y 2 2. ln a3 b How do I solve a natural log equation? First, change it to exponential form Examples: Solve for x 1. ln (3x + 5)2 = 4 2. ln x = 0.1 3. ln (3x - 9) = 21 4. ln ( x2 ) = 12 3 Solving an Exponential Equation with e : You can use natural logs! Examples: Use natural logs to solve each equation: 2x 1. 7e2x + 2.5 = 20 2. ex+1 = 30 p. 481 #1-9 odd, 10, 14-22 even, 23-28 all 3. e 5 7.2 9.1