1. 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved

2. Natural Logarithms Natural Logarithm: if x is a positive real number, then the natural logarithm of x is denoted by log e x or ln x A function given by f(x) = a + ln bx is called a natural logarithm function. Example: Use your calculator to find ln 3 Your scientific calculator has a natural logarithm key on it. ln 3 = 1.0986 Let’s look at the graph of a natural logarithm function...

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

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

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

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

7. 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

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 The line x = 1 is the vertical asymptote of The function.

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