1. **State the problem:** Given the equations $5^m = 35^n$ and $7^p = 35^n$, show that $$n = \frac{mp}{m+p}$$ where $m + p \neq 0$. 2. **Rewrite the given equations:** From $5^m = 35^n$, take the natural logarithm of both sides: $$m \ln 5 = n \ln 35$$ Similarly, from $7^p = 35^n$: $$p \ln 7 = n \ln 35$$ 3. **Express $n$ from both equations:** From the first: $$n = \frac{m \ln 5}{\ln 35}$$ From the second: $$n = \frac{p \ln 7}{\ln 35}$$ 4. **Set the two expressions for $n$ equal:** $$\frac{m \ln 5}{\ln 35} = \frac{p \ln 7}{\ln 35}$$ Multiply both sides by $\ln 35$ (nonzero): $$m \ln 5 = p \ln 7$$ 5. **Rewrite $\ln 35$ using prime factorization:** Since $35 = 5 \times 7$, $$\ln 35 = \ln 5 + \ln 7$$ 6. **Substitute $n$ back using the first expression:** $$n = \frac{m \ln 5}{\ln 5 + \ln 7}$$ Similarly, using the second expression: $$n = \frac{p \ln 7}{\ln 5 + \ln 7}$$ 7. **Use the equality $m \ln 5 = p \ln 7$ to express $\ln 5$ and $\ln 7$ in terms of $m$ and $p$:** Divide both sides by $m p$: $$\frac{\ln 5}{p} = \frac{\ln 7}{m}$$ Let this common value be $k$: $$\ln 5 = k p, \quad \ln 7 = k m$$ 8. **Substitute into $n$ expression:** $$n = \frac{m \ln 5}{\ln 5 + \ln 7} = \frac{m (k p)}{k p + k m} = \frac{m p k}{k (m + p)} = \frac{m p}{m + p}$$ 9. **Conclusion:** We have shown that $$n = \frac{m p}{m + p}$$ provided $m + p \neq 0$ to avoid division by zero. **Final answer:** $$n = \frac{m p}{m + p}$$