How do I solve the following question. 'Find the values of x such that 2log3(x) - log3(x-2) = 2'.

How do I solve the following question. 'Find the values of x such that 2log₃(x) - log₃(x-2) = 2'.To simplify this equation in order to solve it, the first step is to simplify the 2log₃x term. The log rule 'n.log_b(m) = log_b(m)ⁿ' will enable us to do this. Using this rule makes the equation: log₃(x²) - log₃(x-2) = 2. The next step is to recognise that the equation can be simplified by using another log rule 'log_b(m) - log_b(n) = log_b(m/n)'. So by applying this rule to our equation we end up with log₃(x²/(x-2)) = 2. Our equation is now made up of just two terms. Logarithms are just another way of writing indices. If a = b^cthen c = log_ba . In our equation a = (x²/(x-2)), b = 3 and c = 2, by substituting these values into a = b^c, you get (x²/(x-2)) = 3².Now this expression can be rearranged into a quadratic equation, which can then be solved. The steps for this are shown below:(x²/(x-2)) = 9 x² = 9(x-2) x²= 9x -18 x²- 9x + 18 = 0 x²- 9x +18 = 0 (x-6)(x-3)=0 x=3, x=6