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Natural Logarithms

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**If y=ex, then ln y =x. The number e≈2.71828.**

The function y=ex has an inverse, the natural logarithmic function. If y=ex, then ln y =x.

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**8 Simplifying Natural Logarithms Ex. Write as a single natural log.**

3 ln 6 - ln 8 ln 63 - ln 8 ln 63 Use the Power Property Use the Quotient Property Enter 63/8 into the calculator 8 ln 27

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**Ex. Write as a single natural log. 5ln2-ln4 3lnx+lny 4ln3+4lnx**

ln x3y ln 81x4

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**(3x+5)2=e4 (3x+5)2≈54.6 3x+5≈7.39 3x+5≈-7.39 x≈0.797 or x≈-4.130**

Solving a Natural Logarithmic Equation Solve each equation ln(3x+5)2=4 Rewrite in exponential form (3x+5)2=e4 Use a calculator e4 (3x+5)2≈54.6 Take the square root of each side, use ± 3x+5≈ x+5≈-7.39 Use a calculator x≈0.797 or x≈-4.130 Solve for x. x+2 3

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Solve each equation lnx=0.1 ln(3x-9)=21 ln( )=12 x=1.105 x= x+2 __ 3 x= x+2 3

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**Solving an Exponential Equation**

Ex. Use natural logs to solve the equations 7e2x+2.5=20 7e2x=17.5 e2x=2.5 2x=ln 2.5 x=.458 Subtract 2.5 from both sides divide both sides by 7 rewrite in log form Divide both sides by 2

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**Ex. Use natural logs to solve the equations**

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