We can solve the given equation by taking the logarithm of both sides. Any base of the logarithm can be used, but we will use the common logarithm (base 10).
log\((0.25)^y\) = log 32
Using the logarithmic identity log\((a^b)\) = b log\((a)\), we get:
y log\((0.25)\) = log 32
Now, we can evaluate log\((0.25)\) using the logarithmic identity log\((a^n)\) = n log\((a)\), as follows:
log\((0.25)\) = log\((\frac{1}{4})\) = log\((4^{-1})\) = -log\((4)\)
We know that log\((10^x)\) = x, so log\((4)\) = log\((10^{0.602})\) \(\approx\) 0.602
Therefore,
y log\((0.25)\) = log 32
y (-log\((4)\)) = log 32
y (-0.602) = log 32
y = \(\frac{\text{log } 32}{-0.602}\)
Using a calculator, we get:
y \(\approx\) -2.5
Therefore, the value of y is approximately -2.5.
Hence, the answer is: y = -\(\frac{5}{2}\).