Math 151 Section 4.3
Logarithms
The logarithm base b of x is defined as:
f ( x)  log b x
b
log b x
and
g ( x)  b x
 x for x  0
and
Domain [log b x]  (0, )
log b x  y if and only if b y  x .
are inverse functions, thus:
log b b x  x for any x .
Range [log b x]  (, )
Rules of Logarithms
Rules of Exponents
brb s  br  s
log b MN  log b M  log b N
br
M
log b
 log b M  log b N
N
b
log b M t  t log b M
s
 br s
( b r ) t  b tr
When the base of the logarithm is the natural number,
e, log e x  ln x
and this is called the
natural logarithm.
When the base is 10, it is not written. That is
log 10 x  log x .
Example: Using logarithm rules, we can rewrite
ln[ A  B x ]  ln A  x ln B
This is very convenient because exponential functions with base greater than 1 grow large rapidly. But
the logarithm of an exponential function is linear and much easier to work with. This is used in the pH
scale and in the Richter scale.
Change of base formula:
log b x 
ln x
ln b
Examples:
1. Evaluate without a calculator:
a)
log 2 8
b)
log 9 3
d)
log 16
log 4
e)
log 16 64
c)
ln 16
ln 4
2. Solve for the unknown:
a)
2 ln x  ln( x  5)  ln 6
b)
e.02 t  1 .5
c)
4 ( 2 x )  128
3. Find the domain and the vertical asymptote(s) of each function:
a)
f ( x )  ln( x  a )
b)
f ( x )  ln( x 2  1)