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Simplifying Logarithmic Expressions

Properties of Logarithms

The properties of logarithms are analogous to the properties of exponents .

log b b = 1

for any b

Since b 1 = b

log b 1 = 0

for any b

Since b 0 = 1

log b 0 is undefined

for all b

Since there is no x for which b x = 0

log b x is undefined

if x is negative

It may seem like log 2 ( 8 ) should equal 3 , since ( 2 ) 3 = 8 .

But on the other hand, log 2 ( 4 ) doesn't mean anything: the equation ( 2 ) x = 4 has no solution.

log b x y = log b x + log b y
Since b m b n = b m + n
log b x y = log b x log b y