Question

Solve the equation log x = 2 log (mx) for x in terms of m.

Answer 1

Hey there :)
You can apply some rules of logs to get this one moving along.
We can deal with the 2 by using this log rule:\[\large\rm \color{\orangered}{b\cdot \log(a)=\log(a^b)}\]

Answer 2

Make sure you apply the 2 to BOTH the m and the x.

Answer 3

Then you'll have something of this form: \(\large\rm \log(\text{stuff})=\log(\text{other stuff})\)
And since the log function is one-to-one, we can determine that the insides must be equal!\[\large\rm stuff=other~stuff\]

Answer 4

another way to do this is, expand the log on the right hand side
\(\log(mx) = \log(m) + \log(x)\)

then solve for \(\log(x)\) and find x later

Answer 5

Ya, lots of fun ways to approach logs :)

Answer 6

and none dumber than this :-)

\[\log x = 2 \log (mx)\]
\[\frac{\log_x x}{log_x b} = 2 \frac{\log_x mx}{log_x b}\]
where b is what you started in
\[1 = 2 \log_x mx\]
\[x^{\frac{1}{2}} = mx \]