https://file.aiquickdraw.com/mathbot/file/17302465156059apl35yg.jpg

To solve the equation

\[
\frac{1}{2} \log_7 x = 3 \log_7 3,
\]

we can follow these steps:

1. **Remove the coefficients from each side:**  
   We can rewrite the equation by isolating \(\log_7 x\) and \(\log_7 3\) terms.
   
   \[
   \log_7 x = 2 \cdot 3 \log_7 3 = 6 \log_7 3.
   \]

2. **Combine the logarithmic terms:**  
   Using the property \(a \log_b c = \log_b (c^a)\), we rewrite \(6 \log_7 3\) as:
   
   \[
   \log_7 x = \log_7 (3^6).
   \]

3. **Simplify \(3^6\):**
   
   \[
   3^6 = 729.
   \]

   So, the equation becomes:
   
   \[
   \log_7 x = \log_7 729.
   \]

4. **Set the arguments equal to each other:**  
   Since the bases and the logarithmic functions are equal, we conclude that:
   
   \[
   x = 729.
   \]

Thus, the solution is:
\[
\text{Solution set: } \{729\}.
\]

Would you like further details or have any questions?

---

Here are 5 related questions for further practice:

1. Solve \(\log_5 (2x) = 3 \log_5 4\).
2. If \(\frac{1}{3} \log_2 x = \log_2 8\), find \(x\).
3. Solve the equation \(\log_7 (x + 1) = 2 \log_7 5\).
4. If \(\log_{10} (5x) = 2 \log_{10} 3\), determine the value of \(x\).
5. Find the solution to \(\log_b (x + 2) = \log_b 9\).

**Tip:** Remember that if \(\log_b A = \log_b B\), then \(A = B\) (assuming \(A\) and \(B\) are within the domain of the logarithmic function).

Solve the following logarithmic equation: \( \frac{1}{2} \log_7 x = 3 \log_7 3 \)

Learn how to solve the logarithmic equation \( \frac{1}{2} \log_7 x = 3 \log_7 3 \) step-by-step. This solution demonstrates properties of logarithms and equating expressions to find that x = 729. Ideal for students in Grades 10-12.

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