Answer:
24
Step-by-step explanation:
We are given the logarithmic expression:
[tex]\displaystyle{\log_a \left(\dfrac{x^3y}{z^4}\right)}[/tex]
We are also given by the problem that:
[tex]\displaystyle{\log_a x = 3, \ \log_a y = 7, \ \log_a z = -2}[/tex]
From the expression, we will simplify it using two properties:
[tex]\displaystyle{\log_a MN = \log_a M + \log_a N}\\\\\displaystyle{\log_a \dfrac{M}{N} = \log_a M - \log_a N}[/tex]
Therefore, apply the properties to simplify:
[tex]\displaystyle{\log_a x^3y - \log_a z^4}\\\\\displaystyle{\log_a x^3 + \log_a y - \log_a z^4}[/tex]
Next, we will use another property to take an exponent as a coefficient:
[tex]\displaystyle{\log_a x^n = n\log_a x}[/tex]
Hence:
[tex]\displaystyle{3\log_a x + \log_a y - 4\log_a z}[/tex]
Substitute what are given in the problem and the answer will be:
[tex]\displaystyle{3(3)+7-4(-2)}\\\\\displaystyle{9+7+8 = 24}[/tex]
Hence, the answer is 24.
