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Logarithm of one Rule

A mathematical rule of determining the value of logarithm of one is called log one rule.


$\large \log_{b}{1} = 0$


The value of log of 1 can be derived mathematically by considering the zero exponent rule.

Apply Power of 0 Rule

$b$ is a literal and $b$ raised to the power of zero is denoted by $b^0$. According to Power of zero rule, the value of $b^0$ is one.

$\large b^0 = 1$

Use Relation of Logarithm with Exponents

Logarithm is the inverse operation of the exponential system. So, express exponential equation in logarithmic form.

$\,\,\, \therefore \,\,\,\,\,\,$ $\large \log_{b}{1} = 0$


The base of logarithmic term can be any quantity. The logarithm of one to any number is always zero.

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

$(2) \,\,\,\,\,\,$ $\log_{2} 1 = 0$

$(3) \,\,\,\,\,\,$ $\log_{10} 1 = 0$

$(4) \,\,\,\,\,\,$ $\log_{\displaystyle e} 1 = 0$

$(5) \,\,\,\,\,\,$ $\log_{5127} 1 = 0$

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