2log5 (4-x )* log2x (4-x)=3log5 (4-x)-log5 (2x)

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To solve this equation, we will first simplify both sides of the equation using the properties of logarithms.

Starting with the left side:
2log5 (4-x) log2x (4-x)
= log5 (4-x)^2 log2x (4-x)
= log5 (4-x)^2 log2 (4-x) / log2 5
= (2 log5 (4-x) log2 (4-x)) / log2 5
= 2 log5 (4-x) * log2 (4-x) / log2 (5)

Now simplifying the right side:
3log5 (4-x) - log5 (2x)
= log5 (4-x)^3 - log5 (2x)
= log5 ((4-x)^3 / 2x)

Setting the two sides equal to each other:
2 log5 (4-x) * log2 (4-x) / log2 (5) = log5 ((4-x)^3 / 2x)

Since the bases on both sides of the equation are the same, we can remove the logarithm and equate the expressions inside:
2 log2 (4-x) / log2 (5) = (4-x)^3 / 2x

Now, simplify further:
2 log2 (4-x) / log2 (5) = (64 - 48x + 12x^2 - x^3) / 2x

Since log2 (5) is a constant, we can simplify:
2 log2 (4-x) = (64 - 48x + 12x^2 - x^3) / x

Now you can expand the left side using logarithmic properties and solve for x to find the value that satisfies the equation.