If `"log"_(10)5 =x, " then log"_(5) 1250` equals to

Options

A. `3-(1)/(x)`
B. `2+(1)/(x)`
C. `3+(1)/(x)`
D. `2-(1)/(x)`

Answer

To solve the problem, we need to find the value of \( \log_5 1250 \) given that \( \log_{10} 5 = x \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
\[ \log_5 1250 = \log_5 (125 \times 10) \]

**Hint**: Break down the number 1250 into its prime factors or simpler components.

2. **Apply the Logarithm Property**:
Using the property \( \log_a (b \times c) = \log_a b + \log_a c \), we can rewrite the expression:
\[ \log_5 1250 = \log_5 125 + \log_5 10 \]

**Hint**: Remember that logarithms can be split into the sum of logarithms when multiplying.

3. **Simplify \( \log_5 125 \)**:
Since \( 125 = 5^3 \), we can use the power rule of logarithms:
\[ \log_5 125 = \log_5 (5^3) = 3 \]

**Hint**: The power rule states that \( \log_a (b^c) = c \cdot \log_a b \).

4. **Express \( \log_5 10 \)**:
We can use the change of base formula:
\[ \log_5 10 = \frac{1}{\log_{10} 5} \]
Given \( \log_{10} 5 = x \), we have:
\[ \log_5 10 = \frac{1}{x} \]

**Hint**: The change of base formula allows you to convert logarithms to different bases.

5. **Combine the Results**:
Now, substitute back into the expression:
\[ \log_5 1250 = 3 + \frac{1}{x} \]

**Hint**: Combine the results from the previous steps to get the final expression.

### Final Answer:
Thus, the value of \( \log_5 1250 \) is:
\[ \log_5 1250 = 3 + \frac{1}{x} \]