Question
6) \( \log _{4}\left(z^{4} \sqrt{x}\right) \)
Ask by Peterson Sherman. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( 4 \log_{4} z + \frac{1}{2} \log_{4} x \).
Solution
To simplify the expression \( \log_{4}\left(z^{4} \sqrt{x}\right) \), we can use the properties of logarithms:
1. **Product Rule**: \( \log_b (A \cdot B) = \log_b A + \log_b B \)
2. **Power Rule**: \( \log_b (A^k) = k \log_b A \)
3. **Square Root as Exponent**: \( \sqrt{x} = x^{1/2} \)
Let's apply these step by step:
1. **Express the Square Root as an Exponent**:
\[
\sqrt{x} = x^{1/2}
\]
So, the original expression becomes:
\[
\log_{4}\left(z^{4} \cdot x^{1/2}\right)
\]
2. **Apply the Product Rule**:
\[
\log_{4}\left(z^{4}\right) + \log_{4}\left(x^{1/2}\right)
\]
3. **Apply the Power Rule to Each Term**:
\[
4 \log_{4} z + \frac{1}{2} \log_{4} x
\]
So, the simplified form of the expression is:
\[
\log_{4}\left(z^{4} \sqrt{x}\right) = 4 \log_{4} z + \frac{1}{2} \log_{4} x
\]
**Final Answer:**
After expanding, the expression is equal to 4 log₄ z plus one half log₄ x. Thus,
4 log₄ z + ½ log₄ x
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Simplify this solution Extra Insights
To simplify \( \log _{4}\left(z^{4} \sqrt{x}\right) \), we can apply the properties of logarithms. First, we can split the logarithm of a product: \[ \log_4(z^4 \sqrt{x}) = \log_4(z^4) + \log_4(\sqrt{x}). \] Next, we simplify each term. The logarithm of a power can be expressed as: \[ \log_4(z^4) = 4 \log_4(z), \] And since \(\sqrt{x} = x^{1/2}\), we have: \[ \log_4(\sqrt{x}) = \log_4(x^{1/2}) = \frac{1}{2} \log_4(x). \] Combining these, we get: \[ \log_4(z^{4} \sqrt{x}) = 4 \log_4(z) + \frac{1}{2} \log_4(x). \] Thus, the expression simplifies to: \[ \boxed{4 \log_4(z) + \frac{1}{2} \log_4(x)}. \]
