Solve for $x$: $\log_{\frac {3}{4}}(\log_{8} (x^2+7))+\log_{\frac {1}{2}} (\log_{\frac {1}{4}} (x^2+7)^{-1})=-2$ Solve for $x$:  $\log_{\frac {3}{4}}(\log_{8} (x^2+7))+\log_{\frac {1}{2}} (\log_{\frac {1}{4}} (x^2+7)^{-1})=-2$
My Attempt:
$$\log_{\frac {3}{4}}(\frac {1}{3} \log_2 (x^2+7))+\log_{\frac {1}{2}} (\frac {1}{2}\log_2 (x^2+7))=-2$$
$$\log_{\frac {3}{4}}(\frac {1}{3} \log_2 (x^2+7))+1+\log_{1}{2}(\frac {1}{2} \log_2(x^2+7))=-1$$
$$\log_\frac {3}{4} (\frac {1}{3} \log_2 (x^2+7))+\log_{\frac {3}{4}} (\dfrac {3}{4})=-(1+\log_{\frac {1}{2}} (\frac {1}{2} \log_2 (x^2+7)))$$
$$\log_{\frac {3}{4}} (\frac {1}{4} \log_2 (x^2+7))=-(\log_{\frac {1}{2}} (\frac {1}{2})+\log_{\frac {1}{2}} (\frac {1}{2} \log_2 (x^2+7)))$$
$$\log_{\frac {3}{4}} (\frac {1}{4} \log_2 (x^2+7))=-\log_{\frac {1}{2}} (\frac {1}{4} \log_2 (x^2+7))$$

I want to do some replacement but it's going to be tricky.
$\log_8 (x^2+7)=a$
$8^a=x^2+7$
$2^{3a}=x^2+7$
$2^{-3a}=(x^2+7)^{-1}$
$(2^{-2})^{\frac{3}{2}a}=(x^2+7)^{-1}$
$\left(\dfrac{1}{4} \right)^{\frac{3}{2}a}=(x^2+7)^{-1}$
$\log_{\frac{1}{4}} (x^2+7)^{-1}=\dfrac{3}{2}a$
The whole thing changes to:
$\log_{\frac{3}{4}} a+\log_{\frac{1}{2}} \dfrac{3}{2}a=-2$
And with the change of base formula (you'll realize the base doesn't matter) then:
$\dfrac{\log a}{\log \dfrac{3}{4}}+\dfrac{\log \dfrac{3}{2}+\log a}{\log \dfrac{1}{2}}=-2$
$\log a \left(\log \dfrac{1}{2}+\log \dfrac{3}{4} \right)+\log \dfrac{3}{4}\log \dfrac{3}{2}=-2 \cdot \log \dfrac{3}{4}\log \dfrac{1}{2}$
$\log a \log \dfrac{3}{8}=\log \dfrac{3}{4}\log \left(\dfrac{1}{2} \right)^{-2}-\log \dfrac{3}{4}\log \dfrac{3}{2}$
$=\log \dfrac{3}{4} \left(\log 4-\log \dfrac{3}{2} \right)=\log \dfrac{3}{4}\log \dfrac{8}{3}$
$\log a=\dfrac{\log \dfrac{3}{4}\log \dfrac{8}{3}}{\log \dfrac{3}{8}}=\dfrac{-1 \cdot \log \dfrac{3}{4}\log \dfrac{3}{8}}{\log \dfrac{3}{8}}$
$=-\log \dfrac{3}{4}=\log \dfrac{4}{3}$
$a=\dfrac{4}{3}$
$8^{\frac{4}{3}}=16=x^2+7$
$x=\pm 3$
I'm tired now. Let me log out.
