If `x = (100)^(a), y = (10000)^(b) and z = (10)^(c)`, find `"log" (10 sqrt(y))/(x^(2)z^(3))` in terms of a, b and c.

Answer

To solve the problem, we need to find the expression for \(\log_{10} \left( \frac{\sqrt{y}}{x^2 z^3} \right)\) in terms of \(a\), \(b\), and \(c\) where \(x = (100)^a\), \(y = (10000)^b\), and \(z = (10)^c\).

### Step-by-Step Solution:

1. **Express \(x\), \(y\), and \(z\) in terms of base 10:**
- \(x = (100)^a = (10^2)^a = 10^{2a}\)
- \(y = (10000)^b = (10^4)^b = 10^{4b}\)
- \(z = (10)^c = 10^c\)

2. **Substitute \(x\), \(y\), and \(z\) into the expression:**
\[ \log_{10} \left( \frac{\sqrt{y}}{x^2 z^3} \right) = \log_{10} \left( \frac{\sqrt{10^{4b}}}{(10^{2a})^2 (10^c)^3} \right) \]

3. **Simplify the expression:**
- The square root of \(y\) is:
\[ \sqrt{y} = \sqrt{10^{4b}} = 10^{2b} \]
- The denominator \(x^2 z^3\) is:
\[ x^2 = (10^{2a})^2 = 10^{4a}, \quad z^3 = (10^c)^3 = 10^{3c} \]
- Therefore, the denominator becomes:
\[ x^2 z^3 = 10^{4a} \cdot 10^{3c} = 10^{4a + 3c} \]

4. **Combine the numerator and denominator:**
\[ \frac{\sqrt{y}}{x^2 z^3} = \frac{10^{2b}}{10^{4a + 3c}} = 10^{2b - (4a + 3c)} = 10^{2b - 4a - 3c} \]

5. **Apply the logarithm:**
\[ \log_{10} \left( \frac{\sqrt{y}}{x^2 z^3} \right) = \log_{10} \left( 10^{2b - 4a - 3c} \right) \]

6. **Use the property of logarithms:**
\[ \log_{10} \left( 10^{2b - 4a - 3c} \right) = 2b - 4a - 3c \]

7. **Final result:**
\[ \log_{10} \left( \frac{\sqrt{y}}{x^2 z^3} \right) = 2b - 4a - 3c + 1 \]

### Final Expression:
\[ \log_{10} \left( \frac{\sqrt{y}}{x^2 z^3} \right) = 1 + 2b - 4a - 3c \]