Rewrite the expression in terms of exponentials and simplify the results.
-$\ln ( \cosh 6 x - \sinh 6 x )$
A) $\ln \left( e ^ { 6 x } - e ^ { - 6 x } \right)$
B) $- 12 x$
C) $- 6 x$
D) $- 6$

Question

Quizplus · Accepted Answer

The expression $\ln ( \cosh 6 x - \sinh 6 x )$ can be rewritten using the definitions of hyperbolic cosine and hyperbolic sine: $\cosh y = \frac{e^y + e^{-y}}{2}$ and $\sinh y = \frac{e^y - e^{-y}}{2}$. Thus, $\cosh 6x - \sinh 6x = \frac{e^{6x} + e^{-6x}}{2} - \frac{e^{6x} - e^{-6x}}{2} = e^{-6x}$. Taking the natural logarithm of this expression gives $\ln(e^{-6x}) = -6x$.

Rewrite the Expression in Terms of Exponentials and Simplify the Results