If the solve the problem
Question:

If $y=\log _{e} x$, then find $\Delta y$ when $x=3$ and $\Delta x=0.03$.

Solution:

We have

$x=3$

$\Delta x=0.03$

$y=\log _{e} x$

For $x=3$,

$y=\log _{e} 3$

Also, $\frac{d y}{d x}=\frac{1}{x}$

$\Rightarrow\left(\frac{d y}{d x}\right)_{x=3}=\frac{1}{3}$

$\Rightarrow \Delta y=d y=\frac{d y}{d x} d x=\frac{1}{3} \times 0.03=0.01$