Math Doubts

Proof of Logarithm of One identity

The logarithm of one to any base number is equal to zero as per the logarithm of one identity.

$\log_{b}{(1)} \,=\, 0$

Now, let’s learn how to prove the logarithm of one rule in algebraic form.

Express Log of One in Algebraic form

Let $b$ be a literal and represents a base quantity in algebraic form. The logarithm of one to base quantity $b$ is written mathematically as follows.

$\log_{b}{(1)}$

Remember the concept of Zero Power Rule

According to the zero power rule, the base quantity raised to the power of zero is equal to one. In this case, the base quantity is denoted by $b$. Therefore, the zero power law can be written in mathematics as follows.

$b^0 \,=\, 1$

Relation of the Logarithms with Exponents

The logarithmic system is an inverse operation of the exponential system. It means the two concepts have mathematical relationship and it can be used to prove this logarithmic identity in mathematics. According to the mathematical relation between the exponents and logarithms.

$1 \,=\, b^0$ $\,\, \Leftrightarrow \,\,$ $\log_{b}{(1)} \,=\, 0$

Therefore, it is proved that the logarithm of one to the base $b$ is equal to zero.

$\therefore \,\,\,$ $\log_{b}{(1)} \,=\, 0$

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