Question

If $\tan h\left(x\right)=\frac{3}{4}$, then the value of $x$ is

• A. $\sqrt{7}$
• B. $-\sqrt{7}$
• C. $\log \left(\sqrt{7}\right)$
• D. $-\log \left(\sqrt{7}\right)$

Answer 1

The correct option is C

$\log \left(\sqrt{7}\right)$

Explanation for the correct option

Given that $\tan h\left(x\right)=\frac{3}{4}$$\Rightarrow x=\tan h^{-1}\left(\frac{3}{4}\right)$

We know that $\tan h^{-1}\left(y\right)=\frac{1}{2}\ln \left(\frac{1+y}{1-y}\right)$

$\begin{array}{ccclc}\therefore & \tan h^{-1}\left(\frac{3}{4}\right)& =& \frac{1}{2}\ln \left(\frac{1+{\displaystyle \frac{3}{4}}}{1-\frac{3}{4}}\right)& \\ & & =& \frac{1}{2}\ln \left(\frac{7}{1}\right)& \\ \Rightarrow & \tan h^{-1}\left(\frac{3}{4}\right)& =& \ln \left(\sqrt{7}\right)& \left[\because \ln \left(a^b\right)=b\ln \left(a\right)\right]\end{array}$

Hence the correct option is (C) i.e. $\log \left(\sqrt{7}\right)$