Find \({\log _2}32\) .

Given that \({\log _2}\left( {\frac{{{{32}^x}}}{{{8^y}}}} \right)\) can be written as \(px + qy\) , find the value of p and of q.

5     A1     N1

[1 mark]

METHOD 1

\({\log _2}\left( {\frac{{{{32}^x}}}{{{8^y}}}} \right) = {\log _2}{32^x} - {\log _2}{8^y}\)     (A1)

\( = x{\log _2}32 - y{\log _2}8\)     (A1)

\({\log _2}8 = 3\)     (A1)

\(p = 5\) , \(q = - 3\) (accept \(5x - 3y\) )     A1      N3 

METHOD 2

\(\frac{{{{32}^x}}}{{{8^y}}} = \frac{{{{({2^5})}^x}}}{{{{({2^3})}^y}}}\)     (A1) 

\( = \frac{{{2^5}^x}}{{{2^3}^y}}\)     (A1)

\( = {2^{5x - 3y}}\)     (A1)

\({\log _2}({2^{5x - 3y}}) = 5x - 3y\)

\(p = 5\) , \(q = - 3\) (accept \(5x - 3y\) )     A1      N3

[4 marks]

Part (a) proved very accessible.

Although many found part (b) accessible as well, a good number of candidates could not complete their way to a final result. Many gave q as a positive value.