Log5 x =2 log5 2 +4 log25 10

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To simplify the expression, we can use the properties of logarithms:

Log5 x = 2 log5 2 + 4 log25 10

Since log5 2 = log25 2 / log25 5, we can rewrite the expression as:

Log5 x = 2(log25 2 / log25 5) + 4(log25 10)

Next, we can simplify further by using the change of base formula for logarithms:

Log5 x = 2(log2 2 / log2 5) + 4(log10 / log5 5^2)

Log5 x = 2(1 / log2 5) + 4(log10 / 2 log5 5)

Log5 x = 2 / log2 5 + 4(log10 / log5 5^2)

Since log5 5^2 = 2log5 5 = 2, we can further simplify:

Log5 x = 2 / log2 5 + 4(log10 / 2)

Log5 x = 2 / log2 5 + 2 log10

Log5 x = 2 / log2 5 + log10^2

Log5 x = 2 / log2 5 + log100

Finally, we can combine the logarithms using the properties of logarithms:

Log5 x = log(100^(2 / log2 5))

Therefore, the simplified expression is:

Log5 x = log(10,000^(1 / log2 5))