Maximizing My Logs!

Algebra Level 2

In the equation

$4 (\log_{10} x)^{2} + (\log_{10} y)^{2} = 1$

where $x$ and $y$ are positive numbers, find the greatest possible value of $x$ in which this is true.

Extension: Could you do the same for $y$ ?

\sqrt[4]{10}

\frac {1} {\sqrt{10}}

\sqrt{10}

None of the Above

1 solution

Ethan Mandelez
Apr 9, 2021

I'm sure there are lots of other ways to do this, but here's my approach:

$x$ is largest when $(\log_{10} y)^{2}$ is smallest , which is $0$ (i.e. when $y$ is equal to $1$ ). Therefore, we have

$4 (\log_{10} x)^{2} = 1$

$\log_{10} x = \pm \dfrac {1} {2}$

The greatest possible value that $\log_{10} x$ can take is $\dfrac {1} {2}$ , therefore in that case, $x = \sqrt {10}$ .

A similar process can be done to do the same for $y$ .

Good Approach...

Zakir Husain - 2 months ago

thank you!

Ethan Mandelez - 2 months ago