The equation
\[
\ln (x+1)-\ln (x)=2
\]
has the solution $x=$
Hint: First use the properties of logarithms, then apply the exponential on both sides.

Answer

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Answer

Final Answer: $x = \boxed{\frac{1}{e^2 - 1}}$

Steps

Step 1 ：Rewrite the equation using the properties of logarithms: $\ln\left(\frac{x+1}{x}\right) = 2$

Step 2 ：Apply the exponential on both sides to get rid of the logarithm: $\frac{x+1}{x} = e^2$

Step 3 ：Solve the equation for x: $x = \frac{1}{e^2 - 1}$

Step 4 ：Final Answer: $x = \boxed{\frac{1}{e^2 - 1}}$