Question

use natural logarithms to solve the equation (e^{4x}=19). select the correct choice below and, if neces a. (x = \frac{ln(19)}{4}) (simplify your answer. do not round) b. the solution is not a real number.

Explanation:

Step1: Apply natural - log to both sides

Take the natural logarithm of both sides of the equation $e^{4x}=19$. Since $\ln(e^{a}) = a$, we have $\ln(e^{4x})=\ln(19)$.

Step2: Simplify the left - hand side

Using the property $\ln(e^{a}) = a$, the left - hand side $\ln(e^{4x})$ simplifies to $4x$. So, $4x=\ln(19)$.

Step3: Solve for x

Divide both sides of the equation $4x=\ln(19)$ by 4. We get $x = \frac{\ln(19)}{4}$.

Answer:

A. $x=\frac{\ln(19)}{4}$