[tex]\bf log_{{ a}}{{ b}}\implies \cfrac{log_{{ c}}{{ b}}}{log_{{ c}}{{ a}}}\\\\
-----------------------------\\\\
thus
\\\\
log_{10}(e)=\cfrac{1}{ln(10)}\iff log_{10}(e)=\cfrac{1}{log_e(10)}
\\\\
\textit{now, let us do the right-hand-side}
\\\\
\cfrac{1}{log_e(10)}\implies \cfrac{1}{\frac{log_{10}(10)}{log_{10}(e)}}\implies \cfrac{\frac{1}{1}}{\frac{log_{10}(10)}{log_{10}(e)}}
\\\\\\
\cfrac{1}{1}\cdot \cfrac{log_{10}(e)}{log_{10}(10)}\implies \cfrac{log_{10}(e)}{1}\implies log_{10}(e)[/tex]
