If $2^x = 5^y = (10)^{-z}, then \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ =

Question

Accepted Answer

To solve the problem, we need to analyze the given equation: $2^x = 5^y = (10)^{-z}$.

Let's break it down step by step:

1. **Understanding the Equation:**
   - We have three expressions equal to each other: $2^x = 5^y = (10)^{-z}$.
   - This implies that there exists a common value, say $k$, such that:
     $$
     2^x = k, \quad 5^y = k, \quad (10)^{-z} = k
     $$

2. **Expressing in Terms of Logarithms:**
   - From $2^x = k$, we have $x = \log_2 k$.
   - From $5^y = k$, we have $y = \log_5 k$.
   - From $(10)^{-z} = k$, we have $-z = \log_{10} k$, thus $z = -\log_{10} k$.

3. **Finding $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$:**
   - We need to find:
     $$
     \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{\log_2 k} + \frac{1}{\log_5 k} + \frac{1}{-\log_{10} k}
     $$
   - Using the change of base formula, we can express these as:
     $$
     \frac{1}{\log_2 k} = \log_k 2, \quad \frac{1}{\log_5 k} = \log_k 5, \quad \frac{1}{-\log_{10} k} = -\log_k 10
     $$
   - Therefore, the expression becomes:
     $$
     \log_k 2 + \log_k 5 - \log_k 10
     $$

4. **Simplifying the Expression:**
   - Using the properties of logarithms, we know:
     $$
     \log_k 2 + \log_k 5 = \log_k (2 \times 5) = \log_k 10
     $$
   - Thus, the expression simplifies to:
     $$
     \log_k 10 - \log_k 10 = 0
     $$

5. **Conclusion:**
   - Therefore, $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$.

This confirms that the correct answer is indeed Option 2, which is 0.

**Explanation of Incorrect Options:**

- **Option 1 ($\frac{7}{32}$):** This option is incorrect because the simplification of the logarithmic expressions does not yield $\frac{7}{32}$. The calculations clearly show that the sum is 0.
  
- **Option 3 ($\frac{7}{16}$):** Similar to Option 1, this value does not result from the simplification of the given expressions. The sum of the reciprocals of the logarithms is 0, not $\frac{7}{16}$.

- **Option 4 ($\frac{7}{48}$):** This option is also incorrect for the same reasons as above. The mathematical operations and simplifications lead to a sum of 0, not $\frac{7}{48}$.

In conclusion, the logical and mathematical reasoning supports Option 2 as the correct answer, which is 0.