To solve the equation $4 \times \log(7^x) = 52$ using the properties of logarithms, follow these steps:

Step 1: Apply the logarithm power rule

The logarithm power rule states that $\log(a^b) = b \times \log(a)$. So, we can rewrite the equation as: $4 \times x \times \log(7) = 52$

Step 2: Isolate the logarithmic term

Next, divide both sides of the equation by 4 to isolate the $x \times \log(7)$ term: $x \times \log(7) = \frac{52}{4}$ $x \times \log(7) = 13$

Step 3: Solve for $x$

Now, divide both sides of the equation by $\log(7)$ to solve for $x$: $x = \frac{13}{\log(7)}$

Step 4: Calculate the value of $x$

Now, compute the value of $x$ using a calculator: $x = \frac{13}{\log(7)} \approx \frac{13}{0.8451} \approx 15.3791$

So, the value of $x$ rounded to four decimal places is $\boxed{15.3791}$.

Would you like more details, or do you have any questions about this process?

Here are five questions that relate to the problem:

1. How does the power rule of logarithms apply to logarithmic equations?
2. What is the significance of the base of the logarithm in solving equations?
3. How would the solution change if the base of the logarithm was different?
4. Can you derive the logarithm power rule starting from the basic definition of logarithms?
5. How can logarithmic equations be solved if they involve multiple logarithms with different bases?

Tip: When solving logarithmic equations, always check if you can simplify the equation by applying logarithm rules such as the product, quotient, and power rules. This often makes the equation easier to solve.