Use the change-of-base identity to rewrite the equation as
2 log(x³) / log(4x) = 5 log(x) / log(2x)
Multiply both sides by log(4x) log(2x), noting that this is only valid if x > 0 and neither x = 1/2 nor x = 1/4 :
2 log(x³) log(2x) = 5 log(x) log(4x)
Bring down the exponent in the first logarithm:
6 log(x) log(2x) = 5 log(x) log(4x)
Move everything to one side and factorize:
6 log(x) log(2x) - 5 log(x) log(4x) = 0
log(x) (6 log(2x) - 5 log(4x)) = 0
Then either
log(x) = 0 or 6 log(2x) - 5 log(4x) = 0
In the first equation, after taking the exponential of both sides, we get
exp(log(x)) = exp(0) → x = 1
In the second equation, we have
6 log(2x) = 5 log(4x)
Expand the logarithms into sums:
6 (log(2) + log(x)) = 5 (log(4) + log(x))
6 log(2) + 6 log(x) = 5 log(4) + 5 log(x)
Simplify:
log(x) = 5 log(4) - 6 log(2)
4 = 2², so
log(x) = 5 log(2²) - 6 log(2)
log(x) = 10 log(2) - 6 log(2)
log(x) = 4 log(2)
log(x) = log(2⁴)
log(x) = log(16)
Take the exponential of both sides to get
exp(log(x)) = exp(log(16)) → x = 16
