$\displaystyle \log_a(3)=q$ & $\displaystyle \log_a(2)=p$ Express $log_a 72$ in terms of p & q

Question

$\displaystyle \log_a(3)=q$ & $\displaystyle \log_a(2)=p$ Express $log_a 72$ in terms of p & q

Currently I have tried nothing as I cannot even figure out where to begin a demonstration kindly will help much

Many thanks :)

Answer 1

$$\log _{ a }{ 72= } \log _{ a }{ { 2 }^{ 3 }\cdot { 3 }^{ 2 }= } \log _{ a }{ { 2 }^{ 3 }+\log _{ a }{ { 3 }^{ 2 }=3\log _{ a }{ 2 } +2\log _{ a }{ 3 } =3p+2q } } $$

Answer 2

$ \log_a72 = \log_a(2^3 \cdot 3^2)\\ =\log_a (2^3) + \log_a (3^2)\\ = 3 \log_a2 + 2\log_a3\\ = 3p + 2q$

Answer 3

Observe the following rules of logarithm:

$$ \log_n ab = \log_n a + \log_n b$$

$$ \log_n a^b =b\log_n a $$

First you need to simplify the logarithms by factoring 72 into multiples of 2 and 3:

\begin{align} \log_a 72 &= \log_a (2^3 \times 3^2) \\ &= \log_a2^3 + \log_a 3^2\\ &= 3\log_a 2 + 2\log_a3\\ &= 3p + 2q\end{align}