natural logarithms. the number e≈2.71828. the function y=e x has an inverse, the natural...

Natural Logarithms

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Natural Logarithms

The number e≈2.71828.The function y=ex has an inverse, the natural logarithmic function.

If y=ex, then ln y =x.

Simplifying Natural Logarithms

Ex. Write as a single natural log.3 ln 6 - ln 8ln 63 - ln 8ln 63

8ln 27

Use the Power Property

Use the Quotient Property

Enter 63/8 into the calculator

Page 4: Natural Logarithms. The number e≈2.71828. The function y=e x has an inverse, the natural logarithmic function. If y=e x, then ln y =x

Ex. Write as a single natural log.5ln2-ln4

3lnx+lny

4ln3+4lnx

ln 4

ln x3y

ln 81x4

Page 5: Natural Logarithms. The number e≈2.71828. The function y=e x has an inverse, the natural logarithmic function. If y=e x, then ln y =x

Solving a Natural Logarithmic Equation

Solve each equationln(3x+5)2=4

x+2

Rewrite in exponential form

Use a calculator e4

Take the square root of each side, use ±

Use a calculator

Solve for x.

(3x+5)2=e4

(3x+5)2≈54.6

3x+5≈7.39 3x+5≈-7.39

x≈0.797 or x≈-4.130

Page 6: Natural Logarithms. The number e≈2.71828. The function y=e x has an inverse, the natural logarithmic function. If y=e x, then ln y =x

Solve each equation

lnx=0.1

ln(3x-9)=21

ln( )=12

x+2

x+23

x=1.105

x=439605247.8

x=488262.37

Page 7: Natural Logarithms. The number e≈2.71828. The function y=e x has an inverse, the natural logarithmic function. If y=e x, then ln y =x

Solving an Exponential EquationEx. Use natural logs to solve the equations7e2x+2.5=207e2x=17.5e2x=2.52x=ln 2.5x=.458

Subtract 2.5 from both sidesdivide both sides by 7rewrite in log formDivide both sides by 2

Page 8: Natural Logarithms. The number e≈2.71828. The function y=e x has an inverse, the natural logarithmic function. If y=e x, then ln y =x

Ex. Use natural logs to solve the equations

ex+1=30

x=2.4

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