1. ## Logarithm problem

This equation has me at loggerheads...

$\displaystyle 6(\log_{x}2 - \log_{4}x) + 7 = 0$

Even a hint would be appreciated!

2. Hello,

Originally Posted by SVXX
This equation has me at loggerheads...

$\displaystyle 6(\log_{x}2 - \log_{4}x) + 7 = 0$

Even a hint would be appreciated!
Note that $\displaystyle \log_n x=\frac{\ln x}{\ln n}$

So $\displaystyle \log_x 2=\frac{\ln 2}{\ln x}$

And $\displaystyle \log_4 x=\frac{\ln x}{\ln 4}$

Multiply by ln(x), and ln(4) if you want to, then susbtitute y=ln x

3. We haven't done natural logarithms in class yet, LOL.
Could I get some further tips for this? Or is there any other way to solve it?

4. Ok...

There is a formula :

$\displaystyle \log_b x=\frac{1}{\log_a b} \cdot \log_a x$ (took it from Wikipedia)

Taking $\displaystyle b=x$ and $\displaystyle a=4$, you get :

$\displaystyle \log_x 2=\frac{1}{\log_4 x} \cdot \log_4 2$

So now you're working on base 4

This time, multiply by $\displaystyle \log_4 x$

5. If I take $\displaystyle \log_{4}x$ as y, it will turn out to be a quadratic equation. So I guess we'll have two answers...

$\displaystyle \log_{x}2 - \log_{4}x = -7/6$
$\displaystyle => \log_{4}2/\_log_{4}x - \log_{4}x = -7/6$
$\displaystyle => 1/2/y - y = -7/6$
$\displaystyle => 1/2y - y = -7/6$
$\displaystyle => 6y*y - 7y - 3 = 0$

The roots for y are 1.5 and 0.3333.
Taking y = 1.5,
$\displaystyle \log_{4}x = 1.5$
4 raised to 3/2 or 1.5 is x, or x = 8.

6. Originally Posted by SVXX
If I take $\displaystyle \log_{4}x$ as y, it will turn out to be a quadratic equation. So I guess we'll have two answers...

$\displaystyle \log_{x}2 - \log_{4}x = -7/6$
$\displaystyle => \log_{4}2/\_log_{4}x - \log_{4}x = -7/6$
$\displaystyle => 1/2/y - y = -7/6$
$\displaystyle => 1/2y - y = -7/6$
$\displaystyle => 6y*y - 7y - 3 = 0$

The roots for y are 1.5 and 0.3333.
Taking y = 1.5,
$\displaystyle \log_{4}x = 1.5$
4 raised to 3/2 or 1.5 is x, or x = 8.