Suppose $$ \log _{\alpha} x=5, \log _{a} y=3 $$, and $$ \log _{a} z=-1 $$ Find the value of the following expression. $$ \log _{a}\left(\frac{x^{2} y}{z^{3}}\right) $$ 16 13 19

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Given the logarithmic expressions:

$$
\log_{\alpha} x = 5, \quad \log_{a} y = 3, \quad \text{and} \quad \log_{a} z = -1
$$

Assuming $$\alpha = a$$ (as typically $$ \alpha $$ and $$ a $$ would represent the same base unless stated otherwise), we can express all logarithms with the same base $$ a $$:

$$
\log_{a} x = 5
$$

We need to evaluate the expression:

$$
\log_{a}\left(\frac{x^{2} y}{z^{3}}\right)
$$

Using logarithmic properties, we can break this down:

$$
\log_{a}\left(\frac{x^{2} y}{z^{3}}\right) = \log_{a}(x^2) + \log_{a}(y) - \log_{a}(z^3)
$$

Applying the power rule ($$\log_{a}(m^n) = n \log_{a} m$$):

$$
= 2 \log_{a} x + \log_{a} y - 3 \log_{a} z
$$

Substituting the given values:

$$
= 2(5) + 3 - 3(-1) = 10 + 3 + 3 = 16
$$

**Answer: 16**
The value of the expression $$ \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) $$ is 16.

Suppose , and
Find the value of the following expression.

16
13
19

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Suppose , and Find the value of the following expression. 16 13 19