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8.5 Natural Logarithms

8.5 Natural Logarithms. Natural Logarithms. Natural Logarithm: a natural log is a log with base e (the Euler Number) log e x or ln x. Natural Logarithms. Example: Use your calculator to find ln 3 Your calculator has a natural logarithm (Ln) key on it. ln 3 = 1.0986.

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8.5 Natural Logarithms

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  1. 8.5 Natural Logarithms

  2. Natural Logarithms Natural Logarithm:a natural log is a log with base e (the Euler Number) loge x or ln x

  3. Natural Logarithms Example: Use your calculator to find ln 3 Your calculator has a natural logarithm (Ln) key on it. ln 3 = 1.0986

  4. Ex. Graph f(x) = 3 – ln (x – 1) (enter it into your calculator exactly like that) x f(x)

  5. Ex. Graph f(x) = 3 – ln (x – 1) 1.25 x f(x)

  6. Ex. Graph f(x) = 3 – ln (x – 1) 1.25 1.5 2 3 4 5 6 x f(x)

  7. Ex. Graph f(x) = 3 – ln (x – 1) 1.25 1.5 2 3 4 5 6 4.38 x f(x)

  8. Ex. Graph f(x) = 3 – ln (x – 1) 1.25 1.5 2 3 4 5 6 4.38 3.69 3 2.31 1.90 1.611.39 x f(x)

  9. Ex. Graph f(x) = 3 – ln (x – 1) 1.25 1.5 2 3 4 5 6 4.38 3.69 3 2.31 1.90 1.611.39 x f(x)

  10. Ex. Graph f(x) = 3 – ln (x – 1) 1.25 1.5 2 3 4 5 6 4.38 3.69 3 2.31 1.90 1.611.39 f(x) = 3 – ln(x – 1) x f(x)

  11. Natural logarithms can be condensed/expanded using the properties of logarithms: Condense the expressions:a. ln 18 – ln 3b. 3ln x + ln y c. = ln (6) = ln (18/3) = ln x3y =ln 41/2 + ln (62/22) = ln (41/2 * 62/22) = ln (2* 36/4) = ln (2*9) = ln (18)

  12. Expand ln (4xy) = ln (4) + ln (x) + ln (y) ln (3x/y) = ln (3) + ln (x) – ln (y) ln (3x1/2y4) = ln (3) + 1/2ln (x) + 4ln (y)

  13. Solve When solving a ln problem treat it just like a log – get a single ln on each side and set the part in parenthesis equal to each other ln (4x) = ln (8) 4x = 8 4 4 x = 2 2ln (x) = ln (36) ln (x2) = ln (36) x2 = 36 ( )1/2 ( )1/2 x = 6

  14. Solve ln x – ln 10 = ln (8) ln (x/10) = ln (8) x/10= 8 x = 80

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