If `log_(10)x=a,log_(10)y=b" and "log_(10)z=c`, then antilog `(pa+qb-rc)=?`

Options

A. `(pxqy)/(rz)`
B. `px+qy-rz`
C. `(x^(p)y^(q))/(z^(r))`
D. `x^(p)y^(q)z^(r)`

Answer

To solve the problem, we need to find the antilog of the expression \( pa + qb - rc \) given that \( \log_{10} x = a \), \( \log_{10} y = b \), and \( \log_{10} z = c \).

### Step-by-step Solution:

1. **Start with the given logarithmic equations:**
\[ \log_{10} x = a, \quad \log_{10} y = b, \quad \log_{10} z = c \]

2. **Express \( x \), \( y \), and \( z \) in terms of antilogs:**
\[ x = 10^a, \quad y = 10^b, \quad z = 10^c \]

3. **Substitute these expressions into the antilog expression:**
We need to find \( \text{antilog}(pa + qb - rc) \).

4. **Rewrite the expression using the properties of logarithms:**
\[ \text{antilog}(pa + qb - rc) = \text{antilog}(p \log_{10} x + q \log_{10} y - r \log_{10} z) \]

5. **Use the property \( p \log_{10} x = \log_{10} (x^p) \):**
\[ = \text{antilog}(\log_{10} (x^p) + \log_{10} (y^q) - \log_{10} (z^r)) \]

6. **Combine the logarithmic terms:**
Using the property \( \log_{10} a + \log_{10} b = \log_{10} (ab) \) and \( \log_{10} a - \log_{10} b = \log_{10} \left(\frac{a}{b}\right) \):
\[ = \text{antilog}(\log_{10} \left(\frac{x^p y^q}{z^r}\right)) \]

7. **Apply the definition of antilog:**
Since \( \text{antilog}(\log_{10} A) = A \):
\[ = \frac{x^p y^q}{z^r} \]

### Final Answer:
Thus, the antilog \( (pa + qb - rc) \) is:
\[ \frac{x^p y^q}{z^r} \]