10^3x-8=2^5-x

Assuming the usual sloppiness with parentheses, we have

10^(3x-8) = 2^(5-x)
(3x-8)log10 = (5-x)log2
(3log10)x - 8log10 = 5log2 - (log2)x
x(log1000+log2) = 5log2+8log10
xlog2000 = log(32*10^8)
x = log(32*10^8)/log(2000)
x ≈ 2.88

Or, working a different way,

10^(3x-8) = 2^(5-x)
10^3x/10^8 = 2^5/2^x
1000^x * 2^x = 2^5*10^8
1000^x = 32*10^8
and so on