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The equation provided is:

\[
\log_4 x - \log_x 16 = \frac{7}{6} - \log_x 8
\]

We are asked to find the product \(x_1 \cdot x_2\) where \(x_1\) and \(x_2\) are the roots that satisfy the equation.

### Step-by-Step Solution

1. **Rewrite the logarithms:**
   - \( \log_4 x = \frac{\log x}{\log 4} = \frac{\log x}{2 \log 2} \) since \(4 = 2^2\).
   - \( \log_x 16 = \frac{\log 16}{\log x} = \frac{4 \log 2}{\log x}\) since \(16 = 2^4\).
   - \( \log_x 8 = \frac{\log 8}{\log x} = \frac{3 \log 2}{\log x}\) since \(8 = 2^3\).

2. **Substitute these into the equation:**

\[
\frac{\log x}{2 \log 2} - \frac{4 \log 2}{\log x} = \frac{7}{6} - \frac{3 \log 2}{\log x}
\]

Now we have an equation involving logarithms. Solving this will give us the values of \(x_1\) and \(x_2\).

Let me calculate the solution for \(x_1 \cdot x_2\).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through the remaining steps to solve the equation manually. Would you like to continue with that? Let me know if you have any questions.

Here are 5 related questions for practice:
1. How do you convert logarithms of different bases?
2. What are the properties of logarithms that can simplify complex equations?
3. How do you solve logarithmic equations with multiple variables?
4. What is the relation between roots and the coefficients in a quadratic equation?
5. How can you use logarithmic identities in different branches of math?

**Tip:** When solving logarithmic equations, always try to reduce the bases of logarithms to a common base to make the calculations easier.

Jika x1 dan x2 memenuhi 4logx - xlog16 = 7/6 - xlog8, nilai x1 · x2 adalah ....

Learn how to solve a logarithmic equation involving log4(x), logx(16), and logx(8) to find the product of roots x1 · x2. This problem demonstrates the use of logarithmic identities and algebraic manipulations, suitable for students in Grades 10-12.

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