Logarithms
Definition
The logarithm of a number to a given, positive base is simply the exponent of the number written as a
power of the base. For example, with a base of ten (the common logarithm), one would have log 10 1  0 ,
because 1  10 0 ; log 10 10   1 , because 10  101 ; log 10 100   2 , because 100  10 2 ; and, to pick a
number at random, log 10 43.7   1.64 , because 43.7  101.64 . Actually, base 10 is not as common as
base e  2.71827  (Euler’s number), the base of the natural logarithms, abbreviated ln.
Graph
natural logarithm
3
2
1
0
0
1
2
3
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5
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10
-1
-2
-3
Properties
The properties of the logarithm follow from its definition. Let x  bu [equivalently u  log b x  ] and
y  b v [ v  log b  y  ].
 

log b b p  p and b logb  p   p ; i.e., logarithmic and exponential functions are inverse to each other.

The logarithm is only defined for positive numbers: x  bu  0 , because b  0 .

The logarithm ranges from   to   . When x  1 , then log b x   0 because u  0 only when
bu  1 . Likewise when 0  x  1 , then log b x   0 .

log b 1  0 , because 1  b0

log b b   1 , because b  b1

log b xy   log b x   log b  y  , because bu bv  bu v

 
 
log b x p  p log b x  , because b u
p
 b pu