We need to evaluate the function \( f(x) = 7 \log_3 x \) at \( x = 34 \).

### Step 1: Check the Domain
The logarithmic function \( \log_3 x \) is defined for \( x > 0 \). Since \( x = 34 \) is positive, it is within the domain.

### Step 2: Compute \( f(x) \)
We need to evaluate \( \log_3 34 \). Since 34 is not a power of 3, we can't compute this logarithm exactly without a calculator, but we can express it as:
\[
\log_3 34 = \frac{\log 34}{\log 3}
\]
where \( \log \) denotes the logarithm to the base 10 (or natural logarithm, depending on the calculator being used).

### Step 3: Use Approximation
Let's approximate this value:
\[
\log 34 \approx 1.5315, \quad \log 3 \approx 0.4771
\]
\[
\log_3 34 = \frac{1.5315}{0.4771} \approx 3.209
\]

### Step 4: Multiply by 7
Now, multiply this result by 7:
\[
f(34) = 7 \times 3.209 \approx 22.463
\]

### Final Answer
The value of the function \( f(x) \) at \( x = 34 \) is approximately \( \boxed{22.463} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How can you derive the exact value of \( \log_3 34 \) without a calculator?
2. What is the significance of the base of a logarithm in real-world applications?
3. How would the value change if \( x = 35 \) instead?
4. What is the inverse function of \( f(x) = 7 \log_3 x \)?
5. How would you graph the function \( f(x) = 7 \log_3 x \) and analyze its behavior?

**Tip:** When working with logarithms, it can be helpful to remember that changing the base is a common and necessary step for solving many problems.

Learn how to evaluate the function f(x) = 7 log3 x at x = 34 step-by-step. Understand logarithmic functions, use of logarithm change of base formula, and approximation techniques. Suitable for advanced high school students and those interested in logarithmic functions.

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