Logarithms
The rules of logarithms  log_{10} 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
10^{3} = 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
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 
log_{2}= 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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