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Logarithms

The rules of logarithms - log10 and loge 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

ay = x                                   (1)

can be expressed as the "base a logarithm of x " as

loga(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...)

ey = x

where

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

Base e logarithm of x can be expressed as

loge(x) = ln(x) = y

Logarithmic equations log and ln graph

Logarithms
SystemLog to the base ofTerminology
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

loga(x y) = loga(x) + loga(y)                     (2)

loga(x / y) = loga(x) - loga(y)                   (3)

loga(xp) = p loga(x)                        (4)

loga(1 / x) = - loga(x)                        (5)

loga(b) = 1                        (6)

loga(1) = 0                        (7)

loga(0) = undefined                          (8)

loga(x < 0) = undefined                       (9)

loga(x) = logc(x) / logc(a)                     (10)

loga(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 Loge (x)  for x ranging 1 to 1000

Numbers - ln, log e and log 10

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