How would you simplify the expression 3 log_b(X) + 2 log_b(Y) - ½ log_b(Z)?

To simplify the expression ( 3 \log_b(X) + 2 \log_b(Y) - \frac{1}{2} \log_b(Z) ), we can use the properties of logarithms.

First, recall that the property of logarithms states that:

• ( k \log_b(a) = \log_b(a^k) )

• ( \log_b(a) + \log_b(b) = \log_b(ab) )

• ( \log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right) )

Using these properties, we can rewrite each term:

1. ( 3 \log_b(X) ) can be rewritten as ( \log_b(X^3) ).

2. ( 2 \log_b(Y) ) can be rewritten as ( \log_b(Y^2) ).

3. ( -\frac{1}{2} \log_b(Z) ) can be rewritten as ( \log_b(Z^{-\frac{1}{2}}) = \log_b\left(\frac{1}{\sqrt{Z}}\right) ).

Now, we combine these results: