Logarithm of any nonzero positive number to same base is one.

For example, if $b$ is a number, then the logarithm of $b$ to same number is $1$.

$\log_{\displaystyle b} b = 1$

It is called as law of same base of logarithm and this logarithmic rule can be derived in mathematics from its definition.

$m$ is a number and it is expressed as $x$ multiplicative terms on the basis of another number $b$.

$m = b^{\displaystyle x}$

The number of multiplicative terms of $m$ on the basis of $b$ is $x$.

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

Assume, the value of $x$ is $1$.

$m = b^{\displaystyle 1}$

$\implies \log_{\displaystyle b} m = 1$

In this case, the value of $m$ is equal to the value of $b$ if number of multiplying terms is one.

$m = b$

Now, replace $m$ by $b$.

$\therefore \,\, \log_{\displaystyle b} b = 1$

Similarly, replace $b$ by $m$

$\therefore \,\, \log_{\displaystyle m} m = 1$

It expresses that the logarithm of a number to same base is always $1$ mathematically and this logarithmic identity can be used as a formula in logarithms.

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.