\;\normalsize{3log_a 4 - 2log_a 3 + 5log_a 1 }
To simplify the expression 3log_a 4 - 2log_a 3 + 5log_a 1, we can use the properties of logarithms.
1. The first property states that log_a (mn) = log_a m + log_a n. Using this property, we can separate the terms:
3log_a 4 - 2log_a 3 + 5log_a 1 = log_a 4^3 - log_a 3^2 + log_a 1^5
2. The second property states that log_a (m^n) = n * log_a m. Applying this property, we can simplify the exponent terms:
log_a 4^3 - log_a 3^2 + log_a 1^5 = log_a (4^3) - log_a (3^2) + log_a (1^5)
= log_a 64 - log_a 9 + log_a 1
3. The third property states that log_a a = 1. Applying this property, we can further simplify:
log_a 64 - log_a 9 + log_a 1 = 3 - 2 + log_a 1
= 1 + log_a 1
4. The fourth property states that log_a 1 = 0. Applying this property, we get:
1 + log_a 1 = 1 + 0 = 1
Therefore, the simplified expression is 1.