Given expression: [tex]\log _2\left(\log _5\left(x\right)\right)=3[/tex].
[tex]\mathrm{Apply\:log\:rule}:\quad \:a=\log _b\left(b^a\right)[/tex]
[tex]3=\log _2\left(2^3\right)=\log _2\left(8\right)[/tex]
[tex]\log _2\left(\log _5\left(x\right)\right)=\log _2\left(8\right)[/tex]
[tex]\mathrm{For\:}\log _2\left(\log _5\left(x\right)\right)=\log _2\left(8\right)\mathrm{,\:\quad solve\:}\log _5\left(x\right)=8[/tex]
[tex]\log _5\left(x\right)=8[/tex]
Converting log to exponential form.
[tex]log_{b} x = a \ \ \ \ =>x = b^a[/tex]
[tex]x = 5^8[/tex]
x=390625.
Therefore, correct option is C: x=390,625.
