log(base 5)((x^2+2x+1)/x^2)

To simplify the expression log(base 5)((x^2+2x+1)/x^2), we can apply the properties of logarithms.

Firstly, we know that log(base a)(b/c) is equivalent to log(base a)(b) - log(base a)(c). We can use this property to split the numerator and denominator of the fraction inside the logarithm separately.

So, we have:
log(base 5)((x^2+2x+1)/x^2) = log(base 5)(x^2+2x+1) - log(base 5)(x^2)

Next, we can simplify each logarithm individually.

For the first logarithm, log(base 5)(x^2+2x+1):
This can be simplified using the property that log(base a)(a^k) = k. So, if we have log(base a)(a^k), the logarithm cancels out and becomes equal to k.

In our case, the base is 5, and the exponent of 5 is 2, so we can simplify log(base 5)(x^2+2x+1) to just 2.

For the second logarithm, log(base 5)(x^2):
Since there are no other logarithms or exponents involved, we cannot simplify this further, so log(base 5)(x^2) remains as it is.

Now, we can rewrite the simplified expression:
log(base 5)((x^2+2x+1)/x^2) = 2 - log(base 5)(x^2)

Therefore, the simplified form of log(base 5)((x^2+2x+1)/x^2) is 2 - log(base 5)(x^2).