Question

If \( \cosh(x - \log 3) = \sinh x \), then \( x = \):

• A. \( \frac{1}{2} \log 3 \)
• B. \( \frac{1}{2} \log 6 \)
• C. \( \frac{1}{2} \log 5 \)
• D. \( \log 3 \)

Hint:

Hyperbolic Equations}
Use definitions of \( \cosh, \sinh \) in exponential form
Substitute candidate values to verify
Logarithmic transformations often simplify the exponentials

Answer 1

The Correct Option is B

Recall identities: \[ \cosh a = \frac{e^a + e^{-a}}{2}, \quad \sinh b = \frac{e^b - e^{-b}}{2} \] So: \[ \frac{e^{x - \log 3} + e^{-(x - \log 3)}}{2} = \frac{e^x - e^{-x}}{2} \] Multiply both sides by 2: \[ e^{x - \log 3} + e^{-(x - \log 3)} = e^x - e^{-x} \] Let \( x = \frac{1}{2} \log 6 \), test both sides match.