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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.

$lo g_{c} a = \frac{lo g _{b} a}{lo g _{b} c}$

This rule is valid for positive values of

a, b,

and

c,

where

b

and

c

are different than

1 .

Proof

Let

x = lo g_{c} a .

Therefore, by the definition of a logarithm, it is known that

a = c^{x} .

x = lo g_{c} a \Leftrightarrow a = c^{x}

By the Reflexive Property of Equality,

lo g_{b} a

is equal to itself.

lo g_{b} a = lo g_{b} a

Substitute

$a = c^{x}$

lo g_{b} a = lo g_{b} c^{x}

$lo g_{b} (a^{m}) = m \cdot lo g_{b} (a)$

lo g_{b} a = x lo g_{b} c

Substitute

$x = l o g_{c} a$

lo g_{b} a = l o g_{c} a lo g_{b} c

DivEqn

$LHS / lo g_{b} c = RHS / lo g_{b} c$

\frac{lo g _{b} a}{lo g _{b} c} = lo g_{c} a

RearrangeEqn

Rearrange equation

lo g_{c} a = \frac{lo g _{b} a}{lo g _{b} c} ✓

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Proof