Logarithm
Log
If b and a are positive real numbers, b ≠ 1 and x is a real number such that bx = a, then x is called the Logarithm or Log of a to the base b.
log b a = x   ⇔   bx = a
b: base, a: argument, x: exponent
If b = 1, then a = 1 and x = Undefined
Examples
Logarithmic FormExponential Form
log28 = 3⇔23 = 8
log381 = 4⇔34 = 81
log10100 = 2⇔102 = 100
log3 (
1 9
) = − 2		⇔3-2 =
1 9
Deduced Logarithms
log b 1 = 0
log b b = 1
log b a =
1 log a b
log b 0 = Undefined
Laws of Logarithm
Product Rule
logbmn   =   logbm + logbn
Quotient Rule
logb (
m n
)   =   logbm − logbn
Power Rule
logb(mn)   =   n logbm
logb( n x )   =  
logbx n
Change of Base Rule
logbm   =  
logam logab
xlogby   =   ylogbx
Special Logarithms
Common Logarithms
Logarithm with base 10 is known as Common Logarithm.
log10(a) = log(a)
Natural Logarithms
Logarithm with base e is known as Natural Logarithm.
loge(a) = ln(a)