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 in which it lies, we can follow these steps:

### Step 1: Understand the logarithmic function
The logarithm \( \log_{10} 3 \) asks the question: "To what power must 10 be raised to obtain 3?"

### Step 2: Estimate the value of \( \log_{10} 3 \)
We know that:
- \( 10^0 = 1 \) (which is less than 3)
- \( 10^1 = 10 \) (which is greater than 3)

Thus, we can conclude that:
\[ 0 < \log_{10} 3 < 1 \]

### Step 3: Narrow down the range
To find a more precise interval, we can check the values of \( 10^{0.4} \) and \( 10^{0.5} \):
- Calculate \( 10^{0.4} \):
\[ 10^{0.4} \approx 2.5119 \quad (\text{using a calculator or logarithm table}) \]
- Calculate \( 10^{0.5} \):
\[ 10^{0.5} = \sqrt{10} \approx 3.1623 \]

Since \( 2.5119 < 3 < 3.1623 \), we can refine our interval:
\[ 0.4 < \log_{10} 3 < 0.5 \]

### Step 4: Convert the interval to fractions
The interval \( (0.4, 0.5) \) can be expressed as:
\[ \left( \frac{2}{5}, \frac{1}{2} \right) \]

### Step 5: Conclusion
Thus, the value of \( \log_{10} 3 \) lies in the interval \( \left( \frac{2}{5}, \frac{1}{2} \right) \).

### Final Answer
The value of \( \log_{10} 3 \) lies in the interval \( \left( \frac{2}{5}, \frac{1}{2} \right) \).

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