If loge 4 = 1.3868, then loge 4.01 =

(c) $1.3893$

Consider the function $y=f(x)=\log _{e} x$.

Let:

$x=4$

$x+\Delta x=4.01$

$\Rightarrow \Delta x=0.01$

For $x=4$

$\begin{aligned} y &=\log _{e} 4=1.3868 \\ y &=\log _{e} x \end{aligned}$

$\Rightarrow \frac{d y}{d x}=\frac{1}{x}$

$\Rightarrow\left(\frac{d y}{d x}\right)_{x=4}=\frac{1}{4}$

$\Rightarrow \Delta y=d y=\frac{d y}{d x} d x=\frac{1}{4} \times 0.01=0.0025$

$\therefore \log _{e} 4.01=y+\Delta y=1.3893$