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

$b$ is a literal number and its exponent is zero. Therefore, $b$ raised to the power of zero is written in mathematics as $b^0$. Mathematically, any number raised to the power of zero is one.

$\large b^0 = 1$

According to the fundamental relation between an exponential function and logarithmic function. The $b$ raised to the power of zero equals to $1$ can be written in logarithmic form.

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

The value of number $5$ raised to the power of zero is one. So, the logarithm of $1$ to the base $5$ is zero.

$\large 5^0 = 1 \Leftrightarrow \log_{5} 1 = 0$

The base of logarithm can be any number. The logarithm of one to any number is equal to zero in mathematics. It is called log one rule.

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

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

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

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

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

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