Question

Please help! Suppose....

Answer 1

24

Explanation:

We are given the logarithmic expression:

`\displaystyle{\log_a \left((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 (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.