Law of Logarithms 4.4 (1)

Question

The logarithm of a product of two numbers is the same as the sum  of the logarithms of these numbers. So log₄(16 · 64) = log₄(16) +  __________

Answer 1

There is a log rule that states that log_a(bc) = log_ab + log_ac. We have log₄(16 * 64), so a = 4, b = 16, and c = 64. Thus log₄(16 * 64) = log₄16 + log₄64. So our answer for the blank is log₄64 = 3 (since 4³ = 64). Also here is the proof for the log rule. Let log_ab = x and log_ac = y. Thus, a^x = b and a^y = c. Now a^x*a^y = b*c, so a^x+y = b*c. If we rearrange with logs, we get log_a(bc) = x + y. However, log_ab = x and log_ac = y. Therefore, log_a(bc) = log_ab + log_ac.

Answer 2

log64 base 4

log(16x64) = log 16 + log64 for base 4

= 2 +3 = 5

Answer 3

log₄(64)