Logarithms
The logarithm (log) is the inverse operation to exponentiation - and the logarithm of a number is the exponent to which the base - another fixed value - must be raised to produce that number.
The expression
a y = x (1)
can be expressed as the "base a logarithm of x " as
log a (x) = y (1b)
where
a = base
x = antilogarithm
y = logarithm (log)
Example - Logarithm with base 10
Since
10 3 = 1000
- then the base 10 logarithm of 1000 can be expressed as
log 10 (1000) = 3
Natural Logarithm - Logarithm with base e (2.7182...)
e y = x
where
e = 2.7182.... - e constant or Euler's number
Base e logarithm of x can be expressed as
log e (x) = ln (x) = y
| System | Log to the base of | Terminology |
|---|---|---|
| log a | a | log to base a |
| log 10 = lg | 10 | common log |
| log e = ln | e = 2.718281828459.. | natural log |
| log 2 = lb | 2 | log to base 2 |
Rules for Logarithmic Calculations
log a (x y) = log a (x) + log a (y) (2)
log a (x / y) = log a (x) - log a (y) (3)
log a (x p ) = p log a (x) (4)
log a (1 / x) = - log a (x) (5)
log a (b) = 1 (6)
log a (1) = 0 (7)
log a (0) = undefined (8)
log a (x < 0) = undefined (9)
log a (x) = log c (x) / log c (a) (10)
log a (x → ∞) = ∞ (11)
Example - Logarithm Product Rule
log 10 ((5) (6)) = log 10 (5) + log 10 (6)
= 0.6990 + 0.7782
= 1.4772
Conversion of Logarithms
lg (x) = lg (e) ln (x)
= 0.434294 ln (x) (12)
ln (x) = lg (x) / lg (e)
= 2.302585 lg (x) (13)
lb (x) = 1.442695 ln (x)
= 3.321928 lg (x) (15)
Log 10 (x) and Log e (x) for x ranging 1 to 1000
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