Final answer:
To simplify log2(16x+64y), we factor out the common factor of 16, use logarithmic properties to separate the terms, and then simplify the constant logarithmic term, resulting in 4 + log2(x + 4y) as the final simplified expression.
Step-by-step explanation:
To simplify the log2(16x+64y), we can use the properties of logarithms to break it down. Since we are dealing with a logarithm to the base 2, and 16 and 64 are powers of 2 (16 = 24 and 64 = 26), we can use these properties to simplify the expression. We can express the logarithm of a sum by using the fact that log(a + b) does not simplify to log(a) + log(b); instead, we have to look for factors that can be taken out of the logarithm as separate terms.
Here, both 16 and 64 can be factored out to obtain a common factor of 16 (since 64 = 4 × 16). The expression becomes:
log2(16(1x + 4y))
We can now take the factor of 16 outside the logarithm using the logarithmic property that log(a × b) = log(a) + log(b), where a > 0 and b > 0:
log2(16) + log2(1x + 4y)
Since log2(16) equals 4 (because 24 = 16), the expression further simplifies to:
4 + log2(x + 4y)
At this point, the logarithmic expression is fully simplified, and we can't simplify further without additional information about x and y.