Question

ln(x+1) - ln x = 2

Answer 1

start by combining logs:
\[\ln\frac{x+1}{x}=2\]
take everything to base e to cancel log.
\[\frac{x+1}{x}=e^2\]
separate fraction.
\[\frac{x}{x}+\frac{1}{x}=e^2\]
x/x = 1 subtract that to the right.
\[\frac{1}{x}=e^2-1\]
multiply both sides by x.
\[1=e^2x-x\]
collect terms again.
\[-e^2x+x+1=0\]
factor
\[(1-e^2)x+1=0\]
subtract 1
\[(1-e^2)x=-1\]
divide by constants.
\[\large x = \frac{-1}{1-e^2}\]