Find x

$x^{300}=8^{600}$

$(x^3)^{100}=(8^6)^{100}$

It follows that

$x^3=8^6$

$x^3=(8^2)^3$

Finally,

$x=8^2=$ $\large \color{#D61F06}\boxed{64}$

here it is:

$x^{300}$ = $8^{600}$

or, $x^{300}$ = $(8^2)^{300}$

or, $x$ = $8^2$

or, $x$ = $64$

$\large x^{300}=8^{600}$

$300 \ln (x) = 600 \ln(8) = 600 \ln (2)^3\implies 300\ln (x) = (600)(3)\ln (2)$

$\ln(x) = 6\ln (2)\implies x=2^6= 64$

$\begin{aligned} x^{300} & = 8^{600} \\ x^{300} & = \left(8^2\right)^{300} \\ \implies x & = 8^2 = \boxed{64} \end{aligned}$