Find the value of the following expression.

Ask by Bright Hill. in the United States

Jan 26,2025

Question

Find the value of the following expression. $$ \log _{a}\left(\frac{x^{2} y}{z^{3}}\right) $$

UpStudy AI · Accepted Answer

To simplify the expression $$ \log_{a}\left(\frac{x^{2} y}{z^{3}}\right) $$, we can use the properties of logarithms. Specifically, we'll use the **Quotient Rule** and the **Power Rule**.

1. **Quotient Rule**:
   $$
   \log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N
   $$
   
2. **Power Rule**:
   $$
   \log_a(M^k) = k \cdot \log_a M
   $$
   
Applying these rules step-by-step:

1. **Apply the Quotient Rule** to the fraction inside the logarithm:
   $$
   \log_a\left(\frac{x^{2} y}{z^{3}}\right) = \log_a(x^{2} y) - \log_a(z^{3})
   $$
   
2. **Apply the Product Rule** to $$ \log_a(x^{2} y) $$, which states that $$ \log_a(MN) = \log_a M + \log_a N $$:
   $$
   \log_a(x^{2} y) = \log_a(x^{2}) + \log_a(y)
   $$
   
3. **Apply the Power Rule** to both $$ \log_a(x^{2}) $$ and $$ \log_a(z^{3}) $$:
   $$
   \log_a(x^{2}) = 2 \log_a x \quad \text{and} \quad \log_a(z^{3}) = 3 \log_a z
   $$
   
Putting it all together:
$$
\log_a\left(\frac{x^{2} y}{z^{3}}\right) = 2 \log_a x + \log_a y - 3 \log_a z
$$

**Final Simplified Expression:**
$$
2\,\log_a x + \log_a y - 3\,\log_a z
$$
The simplified expression is $$ 2\,\log_a x + \log_a y - 3\,\log_a z $$.

Find the value of the following expression.

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