Question

Use the Change of Base Formula to show that:

log(e)=1/ln(10).
Any help is greatly appreciated!

Answer 1

`\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}{(log_(10)(10))/(log_(10)(e))}\implies \cfrac{(1)/(1)}{(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)`