The value of `log_(10)3` lies in the interval

Options

A. `(2/5,1/2)`
B. `(0,1/2)`
C. `(0,2/5)`
D. None of these

Answer

To find the value of \( \log_{10} 3 \) and determine the interval it lies in, we can follow these steps:

### Step 1: Identify the bounds for 3
We know that \( 3 \) is greater than \( 2 \) and less than \( 4 \). Therefore, we can write:
\[ 2 < 3 < 4 \]

### Step 2: Take the logarithm of all sides
Taking the logarithm (base 10) of all sides gives us:
\[ \log_{10} 2 < \log_{10} 3 < \log_{10} 4 \]

### Step 3: Express \( \log_{10} 4 \)
We can express \( 4 \) as \( 2^2 \). Thus, we can write:
\[ \log_{10} 4 = \log_{10} (2^2) = 2 \cdot \log_{10} 2 \]

### Step 4: Substitute back into the inequality
Now we can substitute this back into our inequality:
\[ \log_{10} 2 < \log_{10} 3 < 2 \cdot \log_{10} 2 \]

### Step 5: Use known values of logarithms
The standard values for logarithms are:
\[ \log_{10} 2 \approx 0.3010 \]
Thus, we can calculate:
\[ 2 \cdot \log_{10} 2 \approx 2 \cdot 0.3010 = 0.6020 \]

### Step 6: Write the final inequality
Now we can write the final inequality:
\[ 0.3010 < \log_{10} 3 < 0.6020 \]

### Conclusion
Therefore, the value of \( \log_{10} 3 \) lies in the interval:
\[ (0.3010, 0.6020) \]