Solve exactly.
\[
\begin{array}{l}
\frac{581 e^{x}}{e^{x}+60}=514 \\
x=
\end{array}
\]

Entry tip: If you include logarithms in your answer, they must be either common logarithms (log) or natural logarithms (ln).

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Step 1 ：Given the equation \(\frac{581 e^{x}}{e^{x}+60}=514\)

Step 2 ：First, we multiply both sides of the equation by \(e^{x}+60\) to get rid of the fraction, which gives us \(581 e^{x} = 514(e^{x}+60)\)

Step 3 ：Then, we simplify the equation to get \(581 e^{x} = 514 e^{x} + 30840\)

Step 4 ：Subtract \(514 e^{x}\) from both sides to get \(67 e^{x} = 30840\)

Step 5 ：Divide both sides by 67 to isolate \(e^{x}\), which gives us \(e^{x} = \frac{30840}{67}\)

Step 6 ：Finally, we take the natural logarithm of both sides to solve for x, which gives us \(x = \ln\left(\frac{30840}{67}\right)\)

Step 7 ：\(\boxed{x = \ln\left(\frac{30840}{67}\right)}\) is the solution to the equation