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Question: The value of the expression \(\log \left( 3+2\sqrt{2} \right)=\) A. \({{\sinh }^{-1}}3\) B. \({...

The value of the expression log⁡(3+22)=\log \left( 3+2\sqrt{2} \right)=
A. sinh⁡−13{{\sinh }^{-1}}3
B. cosh⁡−13{{\cosh }^{-1}}3
C. tanh⁡−13{{\tanh }^{-1}}3
D. coth⁡−13{{\coth }^{-1}}3

Explanation

Solution

In order to solve this problem, we must require the four inverse hyperbolic function formulae which are,
sinh⁡−1x=ln⁡(x+x2+1){{\sinh }^{-1}}x=\ln \left( x+\sqrt{{{x}^{2}}+1} \right)
cosh⁡−1x=ln⁡(x+x2−1){{\cosh }^{-1}}x=\ln \left( x+\sqrt{{{x}^{2}}-1} \right)
tanh⁡−1x=12ln⁡1−x1+x{{\tanh }^{-1}}x=\dfrac{1}{2}\ln \dfrac{1-x}{1+x}
coth⁡−1x=12ln⁡1+x1−x{{\coth }^{-1}}x=\dfrac{1}{2}\ln \dfrac{1+x}{1-x}
We then write log⁡(3+22)\log \left( 3+2\sqrt{2} \right) after manipulating it as log⁡(3+32−1)\log \left( 3+\sqrt{{{3}^{2}}-1} \right) and then compare this form with the four formulae above, and then find the correct option

Complete step by step solution:
The basic inverse hyperbolic functions are sinh⁡−1x,cosh⁡−1x,tanh⁡−1x,coth⁡−1x{{\sinh }^{-1}}x,{{\cosh }^{-1}}x,{{\tanh }^{-1}}x,{{\coth }^{-1}}x . The formulae for these inverse hyperbolic functions are,
sinh⁡−1x=ln⁡(x+x2+1){{\sinh }^{-1}}x=\ln \left( x+\sqrt{{{x}^{2}}+1} \right)
cosh⁡−1x=ln⁡(x+x2−1){{\cosh }^{-1}}x=\ln \left( x+\sqrt{{{x}^{2}}-1} \right)
tanh⁡−1x=12ln⁡1−x1+x{{\tanh }^{-1}}x=\dfrac{1}{2}\ln \dfrac{1-x}{1+x}
coth⁡−1x=12ln⁡1+x1−x{{\coth }^{-1}}x=\dfrac{1}{2}\ln \dfrac{1+x}{1-x}
The given expression that we have in this problem is log⁡(3+22)\log \left( 3+2\sqrt{2} \right) . Now, we can write 222\sqrt{2} as 4×2\sqrt{4\times 2} which is nothing but 8\sqrt{8} . Now, we can write 8\sqrt{8} as 9−1\sqrt{9-1} . 99 is nothing but the square of 33 . So, 9−1=32−1\sqrt{9-1}=\sqrt{{{3}^{2}}-1} . So, we can rewrite the entire expression log⁡(3+22)\log \left( 3+2\sqrt{2} \right) as,
⇒log⁡(3+22)=log⁡(3+32−1)\Rightarrow \log \left( 3+2\sqrt{2} \right)=\log \left( 3+\sqrt{{{3}^{2}}-1} \right)
We can find some similarity between the above expression and the four formulae that we have written before. If we compare log⁡(3+32−1)\log \left( 3+\sqrt{{{3}^{2}}-1} \right) and the formula for cosh⁡−1x{{\cosh }^{-1}}x , we can see that there is a lot common between the two forms of expression. If we put x as 33 in the formula for cosh⁡−1x{{\cosh }^{-1}}x , we get,
cosh⁡−13=ln⁡(3+32−1){{\cosh }^{-1}}3=\ln \left( 3+\sqrt{{{3}^{2}}-1} \right)
This is nothing but the expression that we had derived.
Thus, we can conclude that the value of log⁡(3+22)\log \left( 3+2\sqrt{2} \right) is cosh⁡−13{{\cosh }^{-1}}3 , which is option B.

Note: In order to solve these types of problems, we must remember the four basic inverse hyperbolic functions formulae. This helps us in solving the problems quickly, or we would have to derive the formula for them which would become tedious. We can also find the value of log⁡(3+22)\log \left( 3+2\sqrt{2} \right) using a calculator and compare the value with the values of sinh⁡−13,cosh⁡−13,tanh⁡−13,coth⁡−13{{\sinh }^{-1}}3,{{\cosh }^{-1}}3,{{\tanh }^{-1}}3,{{\coth }^{-1}}3 , found with a calculator.