5log√5 (x)-log5 x=18

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To solve this logarithmic equation, we can first use the properties of logarithms to simplify it.

Starting with the given equation:

5log√5 (x) - log5 x = 18

We can simplify log√5 (x) as log5 (x^(1/2)), since the square root of 5 can be written as 5^(1/2). Applying the power rule of logarithms, we get:

log5 (x^(1/2)) = log5 (√x) = (1/2) * log5 x

Therefore, our equation becomes:

5 (1/2) log5 x - log5 x = 18

Simplify further:

(5/2) * log5 x - log5 x = 18

Combine the logarithms by finding a common denominator:

(5/2 - 2/2) * log5 x = 18

(3/2) * log5 x = 18

Multiply both sides by 2/3 to isolate log5 x:

log5 x = 36

Now, we can rewrite this logarithmic equation in exponential form:

5^(log5 x) = 5^36

x = 5^36

Therefore, the solution to the given equation is x = 5^36.