Logarithms

The rules of logarithms - log₁₀ and log_e for numbers ranging 1 to 1000.

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

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

Logarithms
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
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

Numbers - ln, log e and log 10

Logarithms