To prove that \( z = \frac{xy}{x+y} \) given that \( 3^x = 4^y = 12^z \), we can follow these steps:

### Step 1: Express the equality in logarithmic form
Since \( 3^x = 4^y = 12^z \), let's denote this common value as \( k \). Therefore, we have:
\[
k = 3^x = 4^y = 12^z
\]
Taking the natural logarithm (or logarithm to any base) of all sides, we get:
\[
\ln(k) = x \ln(3) = y \ln(4) = z \ln(12)
\]

### Step 2: Express \( x \), \( y \), and \( z \) in terms of \( \ln(k) \)
From the logarithmic expressions, we have:
\[
x = \frac{\ln(k)}{\ln(3)}, \quad y = \frac{\ln(k)}{\ln(4)}, \quad z = \frac{\ln(k)}{\ln(12)}
\]

### Step 3: Use the relationship between \( \ln(12) \), \( \ln(3) \), and \( \ln(4) \)
Recall that \( 12 = 3 \times 4 \), so:
\[
\ln(12) = \ln(3 \times 4) = \ln(3) + \ln(4)
\]
Thus, we can express \( z \) as:
\[
z = \frac{\ln(k)}{\ln(12)} = \frac{\ln(k)}{\ln(3) + \ln(4)}
\]

### Step 4: Express \( z \) in terms of \( x \) and \( y \)
Now, substitute the expressions for \( x \) and \( y \) from Step 2:
\[
z = \frac{\ln(k)}{\ln(3) + \ln(4)} = \frac{\frac{\ln(k)}{\ln(3)} \cdot \frac{\ln(k)}{\ln(4)}}{\frac{\ln(k)}{\ln(3)} + \frac{\ln(k)}{\ln(4)}}
\]
Simplifying further:
\[
z = \frac{xy}{x + y}
\]
Thus, we have proven that:
\[
z = \frac{xy}{x + y}
\]

### Conclusion
This completes the proof that given \( 3^x = 4^y = 12^z \), we indeed have \( z = \frac{xy}{x + y} \).

Would you like more details or have any other questions?

Here are some related questions:
1. How can this method be applied to similar exponential equations?
2. What would change if \( 3^x \) and \( 4^y \) were raised to different powers?
3. Can this proof be extended to cases where more than three bases are involved?
4. How would the proof differ if we used a different logarithm base?
5. What does this relationship say about the symmetry between \( x \), \( y \), and \( z \)?

**Tip:** Whenever you're working with exponential equations, logarithms are powerful tools to bring the exponents down and analyze the relationships.

Understand and prove the relationship z = xy/(x+y) given the equations 3^x = 4^y = 12^z using logarithmic identities. This problem involves advanced concepts in exponential functions and logarithms, suitable for advanced high school students and beyond.

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