Math Doubts

Fundamental Logarithmic identity

Formula

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

Proof

$b$ is a variable and it is multiplied by itself $x$ times to represent their product $m$ in exponential notation.

$b^{\displaystyle x} = m$

According to the definition of logarithm, the same equation can also be written in logarithm form.

$\log_{b} m = x$

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

The value of $x$ is logarithm of $m$ to base $b$. Substitute the value of $x$ in $b^{\displaystyle x} = m$ equation to obtain the fundamental logarithmic identity.

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

Example

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

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

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

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

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



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