Logarithm equation with a domain check

Practise logarithm equation with a domain check with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

Consider $\ln(x-1)+\ln(x+1)=\ln8$. (a) State the allowed values of x. [1 marks] (b) Solve the equation exactly. [3 marks]

Worked explanation

• Part (a)

  Both logarithm arguments must be positive: $x-1>0$ and $x+1>0$, giving $x>1$.

• Part (b)

  Combine logs: $\ln(x^2-1)=\ln8$, so $x^2=9$. Of $x=\pm3$, only $x=3$ satisfies the domain.

(a) $x>1$.; (b) $x=3$.

Original VectorPaths practice; not an official VCAA question or marking rubric.