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

Step 1: Express the equality in terms of a common variable

Let $k = 3^x = 4^y = 12^z$. Then we have: $k = 3^x = 4^y = 12^z$ Taking the logarithm (in any base, but typically natural logarithms) of all sides gives: $\ln(k) = x \ln(3) = y \ln(4) = z \ln(12)$

Step 2: Solve for $x$, $y$, and $z$ in terms of $\ln(k)$

From the logarithmic expressions: $x = \frac{\ln(k)}{\ln(3)}, \quad y = \frac{\ln(k)}{\ln(4)}, \quad z = \frac{\ln(k)}{\ln(12)}$

Step 3: Use the relationship between $\ln(12)$, $\ln(3)$, and $\ln(4)$

Since $12 = 3 \times 4$, the logarithm of 12 can be written as: $\ln(12) = \ln(3 \times 4) = \ln(3) + \ln(4)$ Thus, we can rewrite $z$ as: $z = \frac{\ln(k)}{\ln(3) + \ln(4)}$

Step 4: Express $z$ in terms of $x$ and $y$

Substitute the expressions for $x$ and $y$ from Step 2: $\frac{1}{z} = \frac{\ln(3) + \ln(4)}{\ln(k)} = \frac{\ln(3)}{\ln(k)} + \frac{\ln(4)}{\ln(k)} = \frac{1}{x} + \frac{1}{y}$

Taking the reciprocal of both sides: $z = \frac{xy}{x + y}$

Conclusion

We have proven that $z = \frac{xy}{x + y}$ given that $3^x = 4^y = 12^z$.

Would you like further details or have any questions?

Here are some related questions:

1. How does this proof change if the base numbers are different?
2. What if we had an equation of the form $a^x = b^y = c^z$?
3. Can this be extended to more complex exponential relationships?
4. What is the significance of the relationship $\ln(12) = \ln(3) + \ln(4)$ in this proof?
5. How can logarithmic identities help in solving other exponential equations?

Tip: When dealing with exponential equations, logarithms can simplify the process of comparing exponents by turning multiplicative relationships into additive ones.