Please help expand the following Logarithmic Expression Acco

Please help expand the following Logarithmic Expression:

According to the solutions manual, the answer is 1/2 + 5/2 log₃x - log₃y however that does not make sense to me. Please help expand the expression and thoroughly explain each step so that I can better understand the answer. Thank you!

use the Laws of Logarithms to expand the expression log, V5 3r5 ogs

Solution

log₃(sqrt(3x^5)/y)

using (log a/b)= log a - log b, we get

log₃(sqrt(3x^5) ) - log ₃y

log₃(3x^5)^1/2- log₃y

using log a^x= x log a, and log (ab)=log a + log b, we get

1/2(log₃3 + log₃x⁵) - log₃y

1/2(1 + 5 log₃x) - log₃y    (since log ₃3=1)

distributing 1/2

1/2 +5/2 log₃x - log₃y