Question

${lim}_{x\to a}\frac{logx-loga}{x-a}$

Answer 1

${lim}_{x\to a}\frac{logx-loga}{x-a}$

Put $x=a+h$ as $h=x-a$

as $x\to a,h\to 0$

$={lim}_{h\to 0}\frac{log(a+h)-loga}{x-a}$

$={lim}_{h\to 0}\frac{1}{h}log\left(\frac{a+h}{a}\right)$

$={lim}_{h\to 0}\frac{1}{a\times \frac{h}{a}}log(1+\frac{h}{a})$

$=\frac{1}{a}{lim}_{h\to 0}\frac{log(1+\frac{h}{a})}{\frac{h}{a}}$

$=\frac{1}{a}\left(1\right)=\frac{1}{a}$