Simplify/express as a single logarithm of an expression.

1/2 log base 3 of p + 2 log base 9 of p

$\begin{array}{rcll} p &=& p \\ 3^{\log_3{(p)}} &=& 9^{\log_9{(p)}} \qquad & | \qquad \log_3{()} \\ \log_3{ ( 3^{\log_3{(p)}} ) }&=& \log_3{ ( 9^{\log_9{(p)}} ) } \\ \log_3{(p)}\cdot \log_3{( 3 )}&=& \log_9{(p)}\cdot \log_3{( 9 )} \qquad & | \qquad \log_3{(3)} = 1\\ \end{array}\\ \boxed{~ \begin{array}{rcll} \log_3{(p)}&=& \log_9{(p)} \cdot \log_3{( 9 )} \qquad & | \qquad \log_3{( 9 )} = \log_3{( 3^2 )} = 2\cdot \log_3{(3)} = 2 \\ \log_3{(p)}&=& \log_9{(p)} \cdot 2 \\ \log_9{(p)} &=& \frac{ \log_3{(p)} }{2} \end{array} ~}$

$\begin{array}{rcll} \frac12 \cdot \log_3{(p)} + 2\cdot \log_9{(p)} &=& \frac12 \cdot \log_3{(p)} + 2\cdot \frac{ \log_3{(p)} }{2} \\ &=& \frac12 \cdot \log_3{(p)} + \log_3{(p)} \\ &=& \frac32 \cdot \log_3{(p)} \\ \end{array}$

(1/2) log₃ p   + (2) log₉ p  =   [ use change of base  to re-write this]

(1/2)[log p] / [log3]   + 2[logp] / [log 9]  =

(1/2) [log p] /[ log 3]  + 2[log p] / [ log(3 * 3)]  =

(1/2) [ log  p] /  [log 3]   +  2 [log p] / [ log3 + log3] =

( [logp] [log3 + log3] + 4[log p] [ log 3] ) / ( 2 log (3)* [2 log3)] ) =

( [log p] [ log3 + log 3 + 4log 3 ] ) / ( 2 log 3 * [ 2 log 3 ])  =

([log p] [ 6 log 3] ) / [ (2log 3) (2log 3)]  =      ......( 6log 3 / 2 log 3   =  3 )

[ log p] [3] / [2 log 3] =

(3/2) [log p / log 3]  =

(3/2) log₃ p