Question

If log_e 4 = 1.3868, then log_e 4.01 =
(a) 1.3968
(b) 1.3898
(c) 1.3893
(d) none of these

Answer 1

(c) 1.3893
$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{function}y=f\left(x\right)={\log }_{e}x. \\ \mathrm{Let}: \\ x=4 \\ x+∆x=4.01 \\ \Rightarrow ∆x=0.01 \\ \mathrm{For}x=4, \\ y=l{\mathrm{og}}_{e}4=1.3868 \\ y={\log }_{e}x \\ \Rightarrow \frac{dy}{dx}=\frac{1}{x} \\ \Rightarrow {\left(\frac{dy}{dx}\right)}_{x=4}=\frac{1}{4} \\ \Rightarrow ∆y=dy=\frac{dy}{dx}dx=\frac{1}{4}\times 0.01=0.0025 \\ \therefore {\log }_{e}4.01=y+∆y=1.3893 \end{gathered}$$