To simplify the expression $\log_3 \sqrt{x} - \log_3 x^4$ as a single logarithm, follow these steps:

1. Rewrite the square root and exponent as powers:

   • $\sqrt{x} = x^{1/2}$, so $\log_3 \sqrt{x} = \log_3 x^{1/2}$.
   • We can also rewrite $x^4$ in the second term directly.

   Now the expression becomes: $\log_3 x^{1/2} - \log_3 x^4$

2. Apply the power rule of logarithms $\log_b a^c = c \cdot \log_b a$:

   • Rewrite each term as: $\frac{1}{2} \log_3 x - 4 \log_3 x$

3. Combine terms:

   • Now, we have: $\frac{1}{2} \log_3 x - 4 \log_3 x = \left( \frac{1}{2} - 4 \right) \log_3 x$

4. Simplify the expression in parentheses:

   • $\frac{1}{2} - 4 = \frac{1}{2} - \frac{8}{2} = -\frac{7}{2}$

   So the expression simplifies to: $-\frac{7}{2} \log_3 x$

Final Answer:

$\log_3 \sqrt{x} - \log_3 x^4 = -\frac{7}{2} \log_3 x$

Would you like a further breakdown, or have any questions?

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Related Practice Questions:

1. Simplify $\log_5 \sqrt{y} - \log_5 y^3$.
2. Express $\log_2 8 + \log_2 16$ as a single logarithm.
3. Rewrite $2 \log_4 x - \log_4 (x + 3)$ as a single logarithm.
4. Simplify $\log_7 (x^5) - \frac{1}{3} \log_7 y$.
5. Combine $\log_9 a + 3 \log_9 b - \log_9 c$ into a single logarithm.

Tip: When combining logarithmic expressions, look for the opportunity to use the power, product, and quotient rules to simplify.