To simplify this logarithmic equation, we can use the property that loga(b) = logc(b) / logc(a), where loga(b) represents the logarithm of b base a.

Given equation: log2x + 6log4x + 9log8x = 14

Using the property mentioned above:
log2x + 6(log4x) + 9(log8x) = 14
log2x + 6(log2^2x) + 9(log2^3x) = 14
log2x + 6(2log2x) + 9(3log2x) = 14
log2x + 12log2x + 27log2x = 14

Combining the like terms:
40log2x = 14

Now we can rewrite the equation in exponential form:
2^(40log2x) = 2^14
x^40 = 2^14
x = (2^14)^(1/40)
x = 2^(14/40)
x = 2^(7/20)

Therefore, x = 2^(7/20) is the solution to the given logarithmic equation.