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.