Exercise 463Algebra › Equations and radicalsHigh schoolAdded 3 Nov 2026

A logarithmic equation

#logarithmic equation#domain#exponential

Statement

Solve in $\mathbb{R}$ the equation
$\ln(x)+\ln(x-3)=\ln(4).$

Hint

Domain: $x>3$ . Combine the logarithms.

Show solutionHide solution

Domain. $x>0$ and $x-3>0$ , so $x>3$ .

Combine. $\ln(x(x-3))=\ln4$ , i.e. $x(x-3)=4$ , i.e. $x^2-3x-4=0$ .

Roots. $(x-4)(x+1)=0$ , so $x=4$ or $x=-1$ .

Check. $x=-1$ is outside the domain. $x=4$ works.

Conclusion. $x=4$ .