The number `log_(20)3` lies in

Options

A. `((1)/(4),(1)/(3))`
B. `((1)/(3),(1)/(2))`
C. `((1)/(2),(3)/(4))`
D. `((3)/(4),(4)/(5))`

Answer

To find the value of \( \log_{20} 3 \), we can use the change of base formula for logarithms, which states:

\[ \log_a b = \frac{\log_c b}{\log_c a} \]

where \( c \) can be any positive number. In this case, we will use base 10 for our calculations.

### Step-by-Step Solution:

1. **Apply the Change of Base Formula**:
\[ \log_{20} 3 = \frac{\log_{10} 3}{\log_{10} 20} \]

2. **Calculate \( \log_{10} 20 \)**:
We can express \( 20 \) as \( 2 \times 10 \). Using the property of logarithms that states \( \log(ab) = \log a + \log b \), we have:
\[ \log_{10} 20 = \log_{10} (2 \times 10) = \log_{10} 2 + \log_{10} 10 \]
Since \( \log_{10} 10 = 1 \), we can write:
\[ \log_{10} 20 = \log_{10} 2 + 1 \]

3. **Use Known Values**:
We can use approximate values for the logarithms:
- \( \log_{10} 2 \approx 0.3010 \)
- \( \log_{10} 3 \approx 0.4771 \)

Thus, we can substitute:
\[ \log_{10} 20 \approx 0.3010 + 1 = 1.3010 \]

4. **Substitute Values into the Formula**:
Now we can substitute the values back into our equation:
\[ \log_{20} 3 = \frac{\log_{10} 3}{\log_{10} 20} = \frac{0.4771}{1.3010} \]

5. **Calculate the Final Value**:
Performing the division:
\[ \log_{20} 3 \approx \frac{0.4771}{1.3010} \approx 0.3661 \]

6. **Determine the Range**:
Since \( 0.3661 \) is between \( 0 \) and \( 1 \), we conclude that:
\[ \log_{20} 3 \text{ lies in the interval } (0, 1) \]