$b^{\displaystyle \log_{b}{m}} \,=\, m$

$m$ is a quantity and it is written in exponential form on the basis of another quantity $b$. The total multiplying factors of $b$ used to obtain the quantity $m$ is $x$.

$m \,=\, \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle x \, factors}$

$\implies m \,=\, b^{\displaystyle x}$

The quantity $m$ is expressed in exponential form as $b^{\displaystyle x}$ and it can be written in logarithmic form on the basis of mathematical relation between exponents and logarithms.

$\log_{b}{m} \,=\, x$

$\implies x \,=\, \log_{b}{m}$

Earlier, it is taken that $m \,=\, b^{\displaystyle x}$

$\implies m \,=\, b^{\displaystyle x}$

$\implies b^{\displaystyle x} \,=\, m$

But it is written in logarithmic form as $x \,=\, \log_{b}{m}$. Now, substitute it in the exponential form equation to get the property of fundamental identity in logarithms.

$\therefore \,\,\,\,\,\, b^{\displaystyle \log_{b}{m}} = m$

Observe the following examples to understand the fundamental rule of logarithms.

$(1) \,\,\,\,\,\,$ $2^{\displaystyle \log_{2}{13}} = 13$

$(2) \,\,\,\,\,\,$ $3^{\displaystyle \log_{3}{5}} = 5$

$(3) \,\,\,\,\,\,$ $4^{\displaystyle \log_{4}{70}} = 70$

$(4) \,\,\,\,\,\,$ $19^{\displaystyle \log_{19}{120}} = 120$

$(5) \,\,\,\,\,\,$ $317^{\displaystyle \log_{317}{1000}} = 1000$

List of most recently solved mathematics problems.

Jul 04, 2018

Limit (Calculus)

Evaluate $\displaystyle \large \lim_{x \,\to\, \tan^{-1}{3}} \normalsize {\dfrac{\tan^2{x}-2\tan{x}-3}{\tan^2{x}-4\tan{x}+3}}$

Jun 23, 2018

Limit (Calculus)

Evaluate $\displaystyle \large \lim_{x \to 0} \normalsize \dfrac{e^{x^2}-\cos{x}}{x^2}$

Jun 22, 2018

Integral Calculus

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

Jun 21, 2018

Limit

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

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.