Find ${\log _3}{p^2}$ .

Find ${\log _3}\left( {\frac{p}{q}} \right)$ .

Find ${\log _3}(9p)$ .

METHOD 1

evidence of correct formula     (M1)

eg   $\log {u^n} = n\log u$ , $2{\log _3}p$

${\log _3}({p^2}) = 12$     A1     N2

METHOD 2

valid method using $p = {3^6}$     (M1)

eg ${\log _3}{({3^6})^2}$ , $\log {3^{12}}$ , $12{\log _3}3$

${\log _3}({p^2}) = 12$     A1     N2

[2 marks]

METHOD 1

evidence of correct formula     (M1)

eg   $\log \left( {\frac{p}{q}} \right) = \log p - \log q$ , $6 - 7$

${\log _3}\left( {\frac{p}{q}} \right) =  - 1$     A1     N2

METHOD 2

valid method using $p = {3^6}$ and $q = {3^7}$     (M1)

eg   ${\log _3}\left( {\frac{{{3^6}}}{{{3^7}}}} \right)$ , $\log {3^{ - 1}}$ , $- {\log _3}3$

${\log _3}\left( {\frac{p}{q}} \right) =  - 1$     A1     N2

[2 marks]

METHOD 1

evidence of correct formula     (M1)

eg   ${\log _3}uv = {\log _3}u + {\log _3}v$ , $\log 9 + \log p$

${\log _3}9 = 2$ (may be seen in expression)     A1

eg   $2 + \log p$

${\log _3}(9p) = 8$     A1     N2

METHOD 2

valid method using $p = {3^6}$     (M1)

eg   ${\log _3}(9 \times {3^6})$ , ${\log _3}({3^2} \times {3^6})$

correct working     A1

eg   ${\log _3}9 + {\log _3}{3^6}$ , ${\log _3}{3^8}$

${\log _3}(9p) = 8$     A1     N2

[3 marks]

Total [7 marks]