I was studying logarithms and got stuck with this problem. Is there anyone who can solve this? Thank you in advance. The question - Given that $p$ and $q$ are positive and that $\ 4(\log_{10} p)^2+2(\log_{10} q)^2=9$ , find the greatest possible value of p such that the equality holds.
Edit
As per asker's note, the answer is:$10\sqrt 10$
The answer is really simple. First you have the equality $4(\log_{10}p)^2+2(\log_{10}q)^2=9$ Then, $9-4(\log_{10}p)^2=2(\log_{10}q)^2$ Now we factor the left hand side, $(3-2(\log_{10}p))(3+2(\log_{10}p))=2(\log_{10}q)^2$. Because the right hand side should always be positive, we deduce that the greatest value of $p$ is when: $2(\log_{10}p)=3$
This is because if $2(\log_{10}p)>3$, then the left hand side is negative. Therefore, $2(\log_{10}p)\leq3$ so 3 is the greatest value that the expression can take:
$\log_{10}p=\frac{1}{2}$
$p=10^{1.5}$