# Change of Base Rule of Logarithms

## Formula

$\large \log_{b}{(m)} = \dfrac{\log_{d}{(m)}}{\log_{d}{(b)}}$

The base of a logarithmic term can be changed mathematically by expressing it as a quotient of two logarithmic terms which contain another quantity as their base. It is usually called as change of base rule and used as a formula in logarithms.

### Proof

The change of base log formula in quotient form is derived in algebraic form on the basis of rules of exponents and also mathematical relation between exponents and logarithms.

#### Basic step

$\log_{b}{m}$ and $\log_{d}{b}$ are two logarithmic terms and it is taken that their values are $x$ and $y$ respectively.

$\log_{b}{m} = x$ and $\log_{d}{b} = y$

Express both logarithmic equations in exponential form by the mathematical relationship between them.

$(1) \,\,\,$ $\log_{b}{m} = x \,\Leftrightarrow\, m = b^{\displaystyle x}$

$(2) \,\,\,$ $\log_{d}{b} \,\,\, = y \,\Leftrightarrow\, b = d^{\displaystyle y}$

#### Changing the Base of Exponential term

Eliminate the base $b$ in the equation $m = b^{\displaystyle x}$ by substituting the $b = d^{\displaystyle y}$.

$\implies$ $m = {(d^{\displaystyle y})}^{\displaystyle x}$

Apply power of power exponents rule to simplify this exponential equation.

$\implies$ $m = d^{\displaystyle xy}$

Express this exponential equation in logarithmic form.

$m = d^{\displaystyle xy} \Longleftrightarrow xy = \log_{d}{m}$

#### Obtaining the property

$\implies$ $xy = \log_{d}{m}$

In fact, $x = \log_{b}{m}$ and $y = \log_{d}{b}$. So, replace them.

$\implies$ $\log_{b}{m} \times \log_{d}{b} = \log_{d}{m}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\log_{b}{m} = \dfrac{\log_{d}{m}}{\log_{d}{b}}$

Thus, the change of base formula is derived in mathematics for changing base of any log term.

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