Use the properties of logarithms to evaluate the given expression in terms of A, B, and/or C?

Let A, B, and C be defined as follows:





log_b_2 = A





log_b_3 = B





log_b_5 = C








Problem: log_b_4th root of 60

Use the properties of logarithms to evaluate the given expression in terms of A, B, and/or C?
1) 4th root of 60 can be written as 60^(1/4) (exponent rules)





2) log_b 60^(1/4) = (1/4) log_b 60 (property of logs)





3) 60 = 2 * 30 = 2 * 2 * 15 = 2^2 * 3 * 5 (factorization)





4) so (1/4) log_b 60 = (1/4) log_b (2^2 * 3 * 5) (substitution)





5) log_b (2^2 * 3 * 5) = log_b 2^2 + log_b 3 + log_b 5 (property of logs)





6) log_b 2^2 = 2 log_b 2 (property of logs, same as #2)





7) putting it all together, we have





log_b 60^(1/4) = (1/4)[2*log_b 2 + log_b 3 + log_b 5]





but log_b 2 = A, log_b 3 = B, log_b 5 = C, so substitute these in to get:





log_b 60^(1/4)


= (1/4)[2*log_b 2 + log_b 3 + log_b 5]


= (1/4)[2*A + B + C]
Reply:problem:log_b_60^(1/4)


this is equal to: 1/4(log_b_60)





60 = 5 x 3 x 2^2


1/4(log_b_(5x3x2^2)) = 1/4(log_b_5 + log_b_3 + log_b_2^2) = 1/4 (log_b_5 + log_b_3 +2*log_b_2) = 1/4 (C + B + 2A)





hope this helps!
Reply:60 = 2^2 * 3 * 5


log[b]( 60^(1/4) ) =


1/4 log[b](60) =


1/4 ( 2A + B + C)