Calculus- Early Transcendentals, 2021a

60 Functions
Since the function f (x)=a x for a ≠ 1 has domain R and range (0,∞), the logarithmic function has domain
(0,∞) and range R. For the most part, we only focus on logarithms with a base larger than 1 (i.e., a > 1)
as these are the most important.
Notice that every logarithm passes through the point (1,0) in the same way that every exponential
function passes through the point (0,1).
Some properties of logarithms are as follows.
Logarithm Properties
Let A,B be positive numbers and b > 0(b ≠ 1) be a base.
•log b (AB)=log b A + log b B,
( ) A
•log b = log
B b A − log b B,
•log b (A n )=nlog b A,wheren is any real number.
Example 2.18: Compute Logarithms
To compute log 2 (24) − log 2 (3) we can do the following:
( ) 24
log 2 (24) − log 2 (3)=log 2 = log
3 2 (8)=3,
since 2 3 = 8.
The Natural Logarithm
As mentioned earlier for exponential functions, the number e ≈ 2.71828...is the most convenient base to
use in Calculus. For this reason we give the logarithm with base e a special name: the natural logarithm.