Question

${\log }_e\frac{(1+3x)}{(1-2x)}$ is equal to

• A. $-5x–\frac{5x^2}{2}–\frac{35x^3}{3}–\dots$
• B. $-5x+\frac{5x^2}{2}–\frac{35x^3}{3}+\dots$
• C. $5x–\frac{5x^2}{2}+\frac{35x^3}{3}–\dots$
• D. $5x+\frac{5x^2}{2}+\frac{35x^3}{3}+\dots$

Answer 1

The correct option is C

$5x–\frac{5x^2}{2}+\frac{35x^3}{3}–\dots$

Explanation for the correct option:

Finding ${\log }_e\frac{(1+3x)}{(1-2x)}$is equal to :

Given : ${\log }_e\frac{(1+3x)}{(1-2x)}$

$$\begin{gathered} {\log }_e\frac{(1+3x)}{(1-2x)}=\log (1+3x)–\log e(1-2x)\mathbf{}\mathbf{[}\mathbf{\because }\mathit{l}\mathit{o}\mathit{g}\mathbf{}\frac{\mathbf{a}}{\mathbf{b}}\mathbf{}\mathbf{=}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{a}\mathbf{}\mathbf{–}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{b}\mathbf{]} \\ \mathbf{ln}\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{x}\mathbf{)}\mathbf{=}\mathbf{x}\mathbf{-}\left(\frac{x^2}{2}\right)\mathbf{+}\left(\frac{x^3}{3}\right)\mathbf{-}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\left(\frac{x^n}{n}\right)\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.} \\ \mathbf{\text{ln(1-x)=-x-}}\left(\frac{x^2}{2}\right)\mathbf{-}\left(\frac{x^3}{3}\right)\mathbf{-}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{-}\left(\frac{x^n}{n}\right)\mathbf{-}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.} \\ \log (1+3x)–\log e(1-2x)=\left(3x-\frac{{\left(3x\right)}^2}{2}+\frac{{\left(3x\right)}^3}{3}-\dots ..\right)-\left(-2x–\frac{{\left(2x\right)}^2}{2}–\frac{{\left(2x\right)}^3}{3}-\dots \right) \\ =\left(3x-\frac{{\left(3x\right)}^2}{2}+\frac{{\left(3x\right)}^3}{3}-\dots ..\right)+\left(2x+\frac{{\left(2x\right)}^2}{2}+\frac{{\left(2x\right)}^3}{3}+\dots \right) \\ =5x–\frac{5x^2}{2}+\frac{35x^3}{3}-\dots \end{gathered}$$

Hence the correct answer is option (C)