Math Doubts

Fundamental Law of Logarithms

Formula

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

Proof

$m$ 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 $x$.

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

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

Express Quantity in Exponential form to Logarithm form

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.

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

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

Replace total factors in Exponential form equation

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.

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

Examples

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

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

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

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

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

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



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

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.

Learn more