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Published byGriffin Christopher Hunter Modified over 4 years ago

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Essential Question: What are some of the similarities and differences between natural and common logarithms.

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In section 8-2, we talked about the number e ≈ 2.71828 being used as a base for exponents. The function e x has an inverse, the natural logarithm function If y = e x, then log e y = x, which is commonly written as ln y = x The properties of common logarithms apply to natural logarithms as well

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Example 1: Simplifying Natural Logarithms ◦ Write 3 ln 6 – ln 8 as a single logarithm 3 ln 6 – ln 8 power rule simplify quotient rule simplify ln 6 3 – ln 8 ln 216 – ln 8 ln 216 / 8 ln 27

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Your Turn ◦ Write each expression as a single logarithm 5 ln 2 – ln 4 3 ln x + ln y ¼ ln 3 + ¼ ln x ln 8 ln x 3 y

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You can use the properties of logarithms to solve natural logarithmic equations Example 3: Solving a Natural Logarithmic Equation ◦ Solve ln (3x + 5) = 4 ln (3x + 5) = 4 Convert to a base of e Subtract 5 from each side Divide each side by 3 e 4 = 3x + 5 54.5982 = 3x + 5 49.5982 = 3x 16.5327 = x

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Your Turn ◦ Solve each equation ln x = 0.1 ln (3x – 9) = 21 1.1052 439,605,247.8277

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You can use natural logarithms to solve exponential equations Example 4: Solving an Exponential Equation ◦ Solve 7e 2x + 2.5 = 20 7e 2x + 2.5 = 20Get the e base by itself Subtract 2.5 from each side Divide each side by 7 Convert to a ln Divide both sides by 2 Use a calculator 7e 2x = 17.5 e 2x = 2.5 ln 2.5 = 2x ln 2.5 / 2 = x 0.4581= x

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Your Turn ◦ Solve each equation e x+1 = 30 2.4012 1.6046

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Page 472-473 ◦ Problems 1 – 9 & 15 – 27, odds ◦ Show your work ◦ Remember to round all problems to 4 decimal places (if necessary)

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