1. 8.5 Natural Logarithms

2. Natural Logarithms Natural Logarithm: a natural log is a log with base e (the Euler Number) log e 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) x f(x) 1.25

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

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

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

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

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

11. Condense the expressions: a. ln 18 – ln 3 b.3ln x + ln y c. = ln ( 18 / 3 ) = ln x 3 y = ln 4 1/2 + ln (6 2 /2 2 ) = ln (4 1/2 * 6 2 /2 2 ) Natural logarithms can be condensed/expanded using the properties of logarithms: = ln (6) = ln (2* 36/4)= ln (2*9)= ln (18)

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

13. Solve ln (4x) = ln (8) 2ln (x) = ln (36) 4x = 8 4 4 x = 2 ln (x 2 ) = ln (36) x 2 = 36 ( ( ) 1/2 x = 6 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

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