Question

Fully expand the logarithm. Simp log₂ (2x⁶/y)

Answer 1

To expand the logarithm log₂(2x⁶/y), we use the properties of logarithms for products and quotients. Using these rules, the expression is simplified to 1 + 6 · log₂(x) - log₂(y).

Step-by-step explanation:

To fully expand the logarithm log2(2x6/y), we use the properties of logarithms. Specifically, we apply two key rules:

1. The logarithm of a product: logb(xy) = logb(x) + logb(y).
2. The logarithm of a quotient: logb(x/y) = logb(x) - logb(y).

Applying these rules, we get:

log2(2x6/y) = log2(2) + log2(x6) - log2(y)

Then we apply the rule of logarithms for an exponent, which states logb(xn) = n · logb(x), giving us:

log2(2x6/y) = log2(2) + 6 · log2(x) - log2(y)

Since log2(2) is simply 1 because 2 is the base of the logarithm, the fully expanded form is:

log2(2x6/y) = 1 + 6 · log2(x) - log2(y)