A rule of changing base of a logarithmic term by expressing it as a product of two logarithmic terms is called the change of base logarithmic rule in product form.

$\large \log_{b} m = \log_{a} m \times \log_{b} a$

It is used as a formula to split a logarithmic term as a product of two logarithmic terms in which one term has different base and second term has the number which is the base of the first term. It is also used in inverse mathematical operation.

The change of base log formula in product form can be derived in algebraic form by considering rules of exponents and the mathematical relation between logarithms and exponents.

$\large \log_{b}{m}$, $\large \log_{a}{m}$ and $\large \log_{b}{a}$ are three logarithmic terms and their values are $x$, $y$ and $z$ respectively. Express three of them in exponential form as per the mathematical relation between exponents and logarithms.

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

$(2) \,\,\,$ $\log_{a}{m} = y \,\Longleftrightarrow\, m = a^{\displaystyle y}$

$(3) \,\,\,$ $\log_{b}{a} \,\,\, = z \,\Longleftrightarrow\, a = b^{\displaystyle z}$

$m = a^{\displaystyle y}$. Change the base of exponential term of this equation on the basis of exponential equation $a = b^{\displaystyle z}$.

$\implies$ $m = {(b^{\displaystyle z})}^{\displaystyle y}$

Use power of power exponents rule to simplify it.

$\implies$ $m = b^{\displaystyle yz}$

Transform this exponential equation in logarithmic form.

$\implies yz = \log_{b}{m}$

Actually, $\log_{a}{m} = y$ and $\log_{b}{a} = z$. Therefore, replace them to obtain the change of base log formula.

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

$\,\,\, \therefore \,\,\,\,\,\,$ $\log_{b}{m} = \log_{a}{m} \times \log_{b}{a}$

List of most recently solved mathematics problems.

Jun 22, 2018

Integral Calculus

Evaluate $\displaystyle \int \dfrac{1+\cos{4x}}{\cot{x}-\tan{x}} dx$

Jun 21, 2018

Limit

Evaluate $\displaystyle \large \lim_{x \to \infty} \normalsize {\sqrt{x^2+x+1}-\sqrt{x^2+1}}$

Jun 20, 2018

Differentiation

Learn how to find derivative of $\sin{(x^2)}$ with respect to $x$.

Jun 19, 2018

Limit (Calculus)

Find $\displaystyle \large \lim_{x \to 0} \normalsize \dfrac{x-\sin{x}}{x^3}$

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.