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Rule

Change of Base Formula

A logarithm of arbitrary base can be rewritten as the quotient of two logarithms with the same base by using the change of base formula.

log_c a= log_b a/log_b c

This rule is valid for positive values of a,b, and c, where b and c are different than 1.

Proof

Let x=log_c a. Therefore, by the definition of a logarithm, it is known that a=c^x. x=log_c a ⇔ a=c^x By the Reflexive Property of Equality, log_b a is equal to itself.

log_b a=log_b a

Substitute

a= c^x

log_b a=log_b c^x

log_b(a^m)= m* log_b(a)

log_b a=xlog_b c

Substitute

x= log_c a

log_b a= log_c alog_b c

DivEqn

.LHS /log_b c.=.RHS /log_b c.

log_b a/log_b c=log_c a