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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)


a = base

x = antilogarithm

y = logarithm (log)

Example - Logarithm with base 10


103 = 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


e = 2.7182.... - e constant or Euler's number

Base e logarithm of x can be expressed as

log e (x) = ln (x) = y

SystemLog to the base ofTerminology
log a a log to base a
log 10 = lg 10 common log
log e = ln e = 2.718281828459.. natural log
log2= 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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