Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $e ^ { x } - e ^ { - x } = 1$
A) $x = \pm \ln \frac { - 1 + \sqrt { 5 } } { 2 } \approx \pm 0.479$
B) $x = \ln \frac { 1 + \sqrt { 5 } } { 2 } \approx 0.481$
C) $x = \ln ( 1 + \sqrt { 5 } ) \approx 0.623$
D) $x = \ln \frac { 1 + \sqrt { 5 } } { 2 } \approx 0.481 ; \ln \frac { 1 - \sqrt { 5 } } { 2 } \approx 0.479$
E) no solution

Question

Quizplus · Accepted Answer

The equation $e^x - e^{-x} = 1$ can be solved by recognizing it as a form of the hyperbolic sine function, $sinh(x) = \frac{e^x - e^{-x}}{2}$. Multiplying both sides of the equation by 2 and rearranging, we get $2sinh(x) = 1$, which implies $x = sinh^{-1}(0.5)$. The inverse hyperbolic sine of 0.5 can be expressed in terms of natural logarithms as $x = \ln(\frac{1 + \sqrt{5}}{2})$, which is approximately 0.481 when rounded to three decimal places. This matches choice B, indicating it is the correct solution.

Find All the Real-Number Roots of the Equation ex−e−x=1e ^ { x } - e ^ { - x } = 1ex−e−x=1

Find All the Real-Number Roots of the Equation $e ^ { x } - e ^ { - x } = 1$