$\displaystyle \log_a(3)=q$ & $\displaystyle \log_a(2)=p$ 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 :)
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Do you know the properties of logarithms? What, for example, is $\log_a 9$ in terms of $p,q$? – lulu Oct 12 '16 at 11:29
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1$$72=2\cdot2\cdot2\cdot3\cdot3$$ – Aakash Kumar Oct 12 '16 at 11:29
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$\log_a 72=\log_a(2^3\cdot 3^2)$ – ajotatxe Oct 12 '16 at 11:29
3 Answers
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$$\log _{ a }{ 72= } \log _{ a }{ { 2 }^{ 3 }\cdot { 3 }^{ 2 }= } \log _{ a }{ { 2 }^{ 3 }+\log _{ a }{ { 3 }^{ 2 }=3\log _{ a }{ 2 } +2\log _{ a }{ 3 } =3p+2q } } $$
haqnatural
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$ \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$
Rohat Kılıç
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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}
John_dydx
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