A rational number which is 50 times its own logarithm to the base 10, is

Options

A. 1
B. 10
C. 100
D. 1000

Answer

To solve the problem, we need to find a rational number \( R \) such that:

\[ R = 50 \cdot \log_{10}(R) \]

### Step 1: Set up the equation
We start with the equation given in the problem:

\[ R = 50 \cdot \log_{10}(R) \]

### Step 2: Rearranging the equation
We can rearrange the equation to isolate the logarithm:

\[ \log_{10}(R) = \frac{R}{50} \]

### Step 3: Multiply both sides by \( \frac{1}{R} \)
To simplify, we multiply both sides by \( \frac{1}{R} \):

\[ \frac{1}{50} = \frac{1}{R} \cdot \log_{10}(R) \]

### Step 4: Use the property of logarithms
Using the property of logarithms, we can express \( \log_{10}(R) \) in another form:

\[ \frac{1}{50} = \log_{10}(R^{\frac{1}{R}}) \]

### Step 5: Exponentiate both sides
Now we exponentiate both sides to eliminate the logarithm:

\[ R^{\frac{1}{R}} = 10^{\frac{1}{50}} \]

### Step 6: Rewrite \( 10^{\frac{1}{50}} \)
To make calculations easier, we can rewrite \( 10^{\frac{1}{50}} \):

\[ R^{\frac{1}{R}} = 10^{\frac{2}{100}} = 100^{\frac{1}{100}} \]

### Step 7: Set the bases equal
Since both sides are in the form \( A^{\frac{1}{A}} \), we can equate the bases:

\[ R = 100 \]

### Conclusion
Thus, the rational number \( R \) that satisfies the condition is:

\[ \boxed{100} \]