8.5 Natural Logarithms

©2001 by R. Villar

All Rights Reserved

Natural Logarithms

Natural Logarithm: if x is a positive real number, then the natural logarithm of x is denoted by

loge 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...

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

1.25

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

1.25 1.5 2 3 4 5 6

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

1.25 1.5 2 3 4 5 64.38

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

1.25 1.5 2 3 4 5 64.38 3.69 3 2.31 1.90 1.611.39

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

1.25 1.5 2 3 4 5 64.38 3.69 3 2.31 1.90 1.611.39

Ex. Graph f(x) = 3 – ln (x – 1)

x

f(x)

1.25 1.5 2 3 4 5 64.38 3.69 3 2.31 1.90 1.611.39

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

Condense the expressions:a. ln 18 – ln 3

b. 3ln x + ln y

c.

= ln 6

= ln x3y

= ln 41/2 + 2 ln 3

= ln 2 + ln 32

= ln 2 + ln 9

= ln 18

12ln4+2(ln6−ln2)

Natural logarithms can be condensed/expanded using the properties of logarithms: