Question

Find ${\log }_e\left(4.04\right)$, if ${\log }_{10}4=0.6021,{\log }_{10}e=0.4343$

Answer 1

Let's assume

$y=lo{g}_e^x$

$x=4$

$y=lo{g}_e^4=\frac{lo{g}_{10}^4}{lo{g}_{10}^e}=\frac{0.6021}{0.4343}=1.3863$

Now $y=lo{g}_e^x$ differentiate w.r.t x

$\frac{dy}{dx}=\frac{1}{x}$

$dy=\frac{1}{x}dx$ $\{\therefore dy\approx △y$, $dx\approx △x$

$△y=\frac{1}{x}△x$ $△x=0.04$

$△y=\frac{1}{4}\times 0.04$ $x=4$

$△y=0.01$

So

$lo{g}_{e}4.04=y+△y$

$=1.3863+0.01$

$=1.3963$