Question

$2\tan h^{-1}\left(\frac{1}{2}\right)$ is equals to

• A. $0$
• B. $\log \left(2\right)$
• C. $\log \left(3\right)$
• D. $\log \left(4\right)$

Answer 1

The correct option is C

$\log \left(3\right)$

Given, $2\tan h^{-1}\left(\frac{1}{2}\right)$

By using formula,

$\tan h^{-1}\left(x\right)=\frac{1}{2}\log \left(\frac{|1+x|}{|1-x|}\right)$

Replacing $x=\frac{1}{2}$

$\Rightarrow \tan h^{-1}\left(\frac{1}{2}\right)=\frac{1}{2}\times \log \left(\frac{1+{\displaystyle \frac{1}{2}}}{1-{\displaystyle \frac{1}{2}}}\right)$

Multiplying both sides by $2$

$\Rightarrow 2\tan h^{-1}\left(\frac{1}{2}\right)=2\times \frac{1}{2}\times \log \left(\frac{1+{\displaystyle \frac{1}{2}}}{1-{\displaystyle \frac{1}{2}}}\right)$

$$\begin{gathered} \Rightarrow 2\tan h^{-1}\left(\frac{1}{2}\right)=\log \left(\frac{{\displaystyle \frac{3}{2}}}{{\displaystyle \frac{1}{2}}}\right) \\ \Rightarrow 2\tan h^{-1}\left(\frac{1}{2}\right)=\log \left(3\right) \end{gathered}$$

Therefore, correct answer is option $\left(c\right)$.