Question

if p=\[\log_{a}2 \], q=\[\log_{a}3 \], r=\[\log_{a}5 \]. Find \[\log_{a}(\frac{ 5\sqrt{3}}{ 3})\] in terms of p, q, r.

Anyone help me please. Don't know where to start.

Thanks.

Answer 1

When you add two logarithms with the same base together, you multiply their insides:
\[\log_{a}2 + \log_{a}3 = \log_{a}(2*3) = \log_{a}6\]
Also when you multiply a number against a logarithm, you put the inside to that power! Behold:
\[3*\log_{a}2 = \log_{a}(2^3) = \log_{a}8\]
...Yet roots are fractional exponents, right? so...
\[(1/2)*\log_{a}3 = \log_{a}(3^{(1/2)}) = \log_{a}(\sqrt{3})\]
So these rules come in handy. Can you put them together to come up with your answer?

Answer 2

\[\log_{a}5+\frac{ 1 }{ 2 } \log_{a}3-\log_{a}2\] so
\[r+\frac{ 1 }{ 2 }q-p\]

Should I write it that?