Question

Please help! Suppose....

Answer 1

Answer:

24

Step-by-step explanation:

We are given the logarithmic expression:

$\displaystyle{\log_a \left(\dfrac{x^3y}{z^4}\right)}$

We are also given by the problem that:

$\displaystyle{\log_a x = 3, \ \log_a y = 7, \ \log_a z = -2}$

From the expression, we will simplify it using two properties:

$\displaystyle{\log_a MN = \log_a M + \log_a N}\\\\\displaystyle{\log_a \dfrac{M}{N} = \log_a M - \log_a N}$

Therefore, apply the properties to simplify:

$\displaystyle{\log_a x^3y - \log_a z^4}\\\\\displaystyle{\log_a x^3 + \log_a y - \log_a z^4}$

Next, we will use another property to take an exponent as a coefficient:

$\displaystyle{\log_a x^n = n\log_a x}$

Hence:

$\displaystyle{3\log_a x + \log_a y - 4\log_a z}$

Substitute what are given in the problem and the answer will be:

$\displaystyle{3(3)+7-4(-2)}\\\\\displaystyle{9+7+8 = 24}$

Hence, the answer is 24.