Question

If log_e 4 = 1.3868, then log_e 4.01 =

Options

• 1.3968

• 1.3898

• 1.3893

• none of these

Answer 1

1.3893

\[\text{ Consider the function } y = f\left( x \right) = \log_e x . \]

\[\text { Let }: \]

\[x = 4\]

\[x + ∆ x = 4 . 01\]

\[ \Rightarrow ∆ x = 0 . 01\]

\[\text { For }x = 4, \]

\[ y = l {og}_e 4 = 1 . 3868\]

\[y = \log_e x\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x}\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 4} = \frac{1}{4}\]

\[ \Rightarrow ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{4} \times 0 . 01 = 0 . 0025\]

\[ \therefore \log_e 4 . 01 = y + ∆ y = 1 . 3893\]