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  4. The approximate value of log e(4.01) is ldots A ldots, if log e 4=1.3863. Here, A refers to

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Q. The approximate value of $\log _e(4.01)$ is $\ldots A \ldots$, if $\log _{ e } 4=1.3863$. Here, $A$ refers to

Application of Derivatives

Solution:

Let $y=f(x)=\log _e x, x=4$
and $ x+\Delta x=4.01$ Then,
$\Delta x=0.01$
Now, for $ x=4$ we have,
$y = f(4) = \log_e 4 = 1.3863$
let $ d x=\Delta x=0.01$
Now, $ y=\log _e x$
$\Rightarrow \frac{d y}{d x}=\frac{1}{x}$
$\Rightarrow \left(\frac{d y}{d x}\right)_{x=4}=\frac{1}{4}$
$\therefore \Delta y=\frac{d y}{d x} \Delta x$
$\Rightarrow \Delta y=\frac{1}{4} \times 0.01$
$\Rightarrow \Delta y=0.0025 (\because d y \approx \Delta y)$
$\therefore \log _e(4.01)=y+\Delta y$
$=1.3863+0.0025$
$=1.3888$