Use the laws of logarithms to expand and simplify the expression.
\[
\ln \left(\frac{e^{x}}{1+e^{x}}\right)
\]
Answer
The expanded and simplified expression is \(\boxed{x - \ln(1+e^{x})}\)
Step 1 ：Rewrite the expression as the difference of two logarithms using the property of logarithms that states the logarithm of a quotient is the difference of the logarithms: \(\ln\left(\frac{e^{x}}{1+e^{x}}\right) = \ln(e^{x}) - \ln(1+e^{x})\)
Step 2 ：Use the property of logarithms that states the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number to bring down the exponent in the first logarithm: \(\ln(e^{x}) - \ln(1+e^{x}) = x - \ln(1+e^{x})\)
Step 3 ：Simplify the expression by recognizing that the natural logarithm of e is 1: \(x - \ln(1+e^{x})\)
Step 4 ：The expanded and simplified expression is \(\boxed{x - \ln(1+e^{x})}\)
