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

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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 

ay = x                                   (1)

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

loga (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

log10 (1000) = 3

Natural Logarithm - Logarithm with base e (2.7182...)

ey = x


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

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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  • Engineering ToolBox, (2009). Logarithms. [online] Available at: [Accessed Day Mo. Year].

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