Question
Solve the equation \(2 – {\log _3}(x + 7) = {\log _{\tfrac{1}{3}}}2x\) .
▶️Answer/Explanation
Markscheme
\({\log _3}\left( {\frac{9}{{x + 7}}} \right) = {\log _3}\frac{1}{{2x}}\) M1M1A1
Note: Award M1 for changing to single base, M1 for incorporating the 2 into a log and A1 for a correct equation with maximum one log expression each side.
\(x + 7 = 18x\) M1
\(x = \frac{7}{{17}}\) A1 [5 marks]
Question
Consider \(a = {\log _2}3 \times {\log _3}4 \times {\log _4}5 \times \ldots \times {\log _{31}}32\). Given that \(a \in \mathbb{Z}\), find the value of a.
▶️Answer/Explanation
Markscheme
\(\frac{{\log 3}}{{\log 2}} \times \frac{{\log 4}}{{\log 3}} \times \ldots \times \frac{{\log 32}}{{\log 31}}\) M1A1
\( = \frac{{\log 32}}{{\log 2}}\) A1
\( = \frac{{5\log 2}}{{\log 2}}\) (M1)
\( = 5\) A1
hence \(a = 5\)
Note: Accept the above if done in a specific base eg \({\log _2}x\).
[5 marks]