Insert integers, decimals, or fractions in the boxes to write the coefficients.
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Mar 25,2025

Question

Insert integers, decimals, or fractions in the boxes to write the coefficients. $$ \log \left(\frac{x^{5}}{y^{8 \cdot \sqrt[4]{z}}}\right)=5 \cdot \log x+ $$ Number $$ \cdot \log y+ $$ 0

UpStudy AI · Accepted Answer

To rewrite the expression $$ \log \left(\frac{x^{5}}{y^{8 \cdot \sqrt[4]{z}}}\right) $$ using the properties of logarithms, we can break it down step by step.

1. **Apply the Quotient Rule of Logarithms**:
   $$
   \log \left(\frac{a}{b}\right) = \log a - \log b
   $$
   Thus,
   $$
   \log \left(\frac{x^{5}}{y^{8 \cdot \sqrt[4]{z}}}\right) = \log(x^{5}) - \log(y^{8 \cdot \sqrt[4]{z}})
   $$

2. **Apply the Power Rule of Logarithms**:
   $$
   \log(a^{b}) = b \cdot \log a
   $$
   Therefore,
   $$
   \log(x^{5}) = 5 \cdot \log x
   $$
   and
   $$
   \log(y^{8 \cdot \sqrt[4]{z}}) = (8 \cdot \sqrt[4]{z}) \cdot \log y
   $$

3. **Combine the Results**:
   Now we can substitute these back into the equation:
   $$
   \log \left(\frac{x^{5}}{y^{8 \cdot \sqrt[4]{z}}}\right) = 5 \cdot \log x - (8 \cdot \sqrt[4]{z}) \cdot \log y
   $$

From this, we can identify the coefficients:
- The coefficient of $$ \log y $$ is $$ -8 \cdot \sqrt[4]{z} $$.
- The constant term is $$ 0 $$.

Thus, the completed expression is:
$$
\log \left(\frac{x^{5}}{y^{8 \cdot \sqrt[4]{z}}}\right) = 5 \cdot \log x - 8 \cdot \sqrt[4]{z} \cdot \log y + 0
$$

So, the coefficients are:
- For $$ \log y $$: $$ -8 \cdot \sqrt[4]{z} $$
- For the constant: $$ 0 $$
The coefficients are: - For $$ \log y $$: $$ -8 \cdot \sqrt[4]{z} $$ - For the constant: $$ 0 $$

Insert integers, decimals, or fractions in the boxes to write the coefficients.
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Insert integers, decimals, or fractions in the boxes to write the coefficients. Number 0