If logb3=A and logb5=C express the logarithms in terms of A and C

Question

logb45=

logb125=

logb1/15

Answer 1

We are given 2 logarithms of prime factors 3 and 5, and all of the questions involve these prime factors. We can use the rules of logarithms to write them in terms of A and C.

45 = 3 • 3 • 5.

log_b 45 =log_b (3 • 3 • 5)

Using the product rule, we can expand this to

log_b 3 + log_b 3 + log_b 5 = A + A + C = 2A + C

125 = 5³

Using the power rule,

log_b 125 = log_b 5³ = 3log_b 5 = 3A

¹/₁₅ = ¹/₃ • ¹/₁₅ = 3^-1 • 5^-1

Here we can use the product rule, and then the power rule

log_b ¹/₁₅ = log_b (¹/₃ • ¹/₁₅ )= log_b (3^-1 • 5^-1) = log_b 3^-1 + log_b 5^-1 = -log_b 3 + -log_b 5

This is equivalent to -A - C