If loga2 = x and loga3 = y, what is the value of loga16?

Concept: The power rule of logarithms states that loga(mn) = n · logam. An exponent inside the argument of a logarithm can be pulled out as a multiplier in front of the log.

Application:

1. Write the argument 16 as a power of 2: 16 = 24.

2. Apply the power rule: loga(24) = 4 · loga2.

3. Substitute the given value loga2 = x, giving 4 · x = 4x.

Cross-check: The value of y (loga3) is not required, since 16 has no factor of 3. Adding the two givens gives loga2 + loga3 = loga6 — that is the log of 6, not of 16 — so x + y is not the answer. The result is 4x.