Equation with the natural logarithm

Text

Solve the equation $\ln x+\ln(x-3)=\ln 4$.

Solution

Existence conditions: $x>0$ and $x-3>0$, i.e. $x>3$. Use the sum rule and injectivity of the logarithm: $\ln\big[x(x-3)\big]=\ln 4\;\Rightarrow\; x(x-3)=4\;\Rightarrow\; x^{2}-3x-4=0.$ The roots are $x=4$ and $x=-1$; only $x=4$ satisfies $x>3$. $\boxed{\,x=4\,}$