I'm having trouble finding the steps to solve for $x$. The solutions to this equation are $x=-4$, $x=-2$, and $x=0.76666$ when solved graphically and through the solve function of a TI-nspire cx CAS.
I tried to isolate $x$ by using various log and power rules, but the result was still something I did not know how to solve.
$$x^2 = \left(\frac 12\right)^x $$
$$\log x^2=\log\left(\frac12\right)^x$$
$$2\log x=x\log\left(\frac12\right)$$
$${\log x \over x}={\log \frac12 \over 2}$$
I also tried the following:
$$x^2 = \left(\frac 12\right)^x $$
$$x^2 = {1^x \over 2^x} $$
$1^n=1$ for all real $n$
$$x^2={1 \over 2^x} $$
$$x=\sqrt {1 \over 2^x} ={1 \over \sqrt {2^x}}={1 \over [2^x]^{1/2}}= {1 \over 2^{x/2}}$$
How do you do this?