Question

The value of \[ \tanh(\log x) \] is

• A. \(\dfrac{x+1}{x-1}\)
• B. \(\dfrac{x^2+1}{x^2-1}\)
• C. \(\dfrac{x^2-1}{x^2+1}\)
• D. \(2x\)

Hint:

For expressions involving \(\log x\) inside hyperbolic functions, substitute \(e^{\log x}=x\) and then simplify using the definitions of hyperbolic functions.

Answer 1

The Correct Option is C

Step 1: Apply the definition \(\tanh t = \tfrac{e^t - e^{-t}}{e^t+e^{-t}}\).
With \(t = \log x\): \(e^{\log x}=x\) and \(e^{-\log x}=1/x\). So \[\tanh(\log x) = \frac{x - 1/x}{x + 1/x} = \frac{x^2-1}{x^2+1}.\]
\[\boxed{\frac{x^2-1}{x^2+1}}\]