Find the slope of the tangent line for f(x) = 7log base 4 (x) at x=2

To find the slope of the tangent line to the function \( f(x) = 7 \log_4(x) \) at \( x = 2 \), we need to first compute the derivative of \( f(x) \), since the slope of the tangent line at any point is given by the value of the derivative at that point.

### Step 1: Differentiate \( f(x) = 7 \log_4(x) \)
We use the change of base formula for logarithms:
\[
\log_b(x) = \frac{\ln(x)}{\ln(b)}
\]
Thus, \( \log_4(x) = \frac{\ln(x)}{\ln(4)} \), and the function becomes:
\[
f(x) = 7 \cdot \frac{\ln(x)}{\ln(4)}
\]

Now, take the derivative of \( f(x) \) with respect to \( x \):
\[
f'(x) = 7 \cdot \frac{1}{\ln(4)} \cdot \frac{1}{x}
\]
Simplifying, we get:
\[
f'(x) = \frac{7}{x \ln(4)}
\]

### Step 2: Evaluate the derivative at \( x = 2 \)
Now, substitute \( x = 2 \) into the derivative to find the slope of the tangent line:
\[
f'(2) = \frac{7}{2 \ln(4)}
\]
Since \( \ln(4) = 2 \ln(2) \), we get:
\[
f'(2) = \frac{7}{2 \cdot 2 \ln(2)} = \frac{7}{4 \ln(2)}
\]

Thus, the slope of the tangent line at \( x = 2 \) is \( \frac{7}{4 \ln(2)} \).

### Final Answer:
The slope of the tangent line at \( x = 2 \) is \( \frac{7}{4 \ln(2)} \).

Would you like more details on the steps? Here are some related questions:

1. What is the derivative of a general logarithmic function?
2. How does the change of base formula for logarithms work?
3. Can you explain the process of finding the tangent line for any function?
4. Why do we use natural logarithms (\( \ln \)) when differentiating logarithms with other bases?
5. How can we apply this method to other logarithmic functions?

**Tip:** When working with logarithms in different bases, it's often helpful to convert to natural logarithms to simplify differentiation.

This detailed solution explains how to find the slope of the tangent line for the logarithmic function f(x) = 7log base 4 (x) at x = 2. The solution covers key calculus concepts, including the derivative of a logarithmic function, the change of base formula, and step-by-step differentiation. Suitable for students in Grades 11-12 or Calculus I learners.

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