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

10³ = 1000

- then the base 10 logarithm of 1000 can be expressed as

log₁₀(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

Logarithms

System	Log to the base of	Terminology
loga	a	log to base a
log10 = lg	10	common log
loge = 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

log₁₀((5) (6)) = log₁₀(5) + log₁₀(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)

Log₁₀ (x) and Log_e (x)  for x ranging 1 to 1000