Prove that ` log_(10) 2" lies between " 1/4 and 1/3`.

Answer

To prove that \( \log_{10} 2 \) lies between \( \frac{1}{4} \) and \( \frac{1}{3} \), we will use the properties of logarithms and exponentiation.

### Step 1: Establish the inequalities
We need to show:
\[ \frac{1}{4} < \log_{10} 2 < \frac{1}{3} \]

### Step 2: Convert the inequalities using exponentiation
Using the definition of logarithms, we can rewrite the inequalities:
1. From \( \log_{10} 2 > \frac{1}{4} \):
\[ 2 > 10^{\frac{1}{4}} \]
2. From \( \log_{10} 2 < \frac{1}{3} \):
\[ 2 < 10^{\frac{1}{3}} \]

### Step 3: Calculate \( 10^{\frac{1}{4}} \) and \( 10^{\frac{1}{3}} \)
Now we need to evaluate \( 10^{\frac{1}{4}} \) and \( 10^{\frac{1}{3}} \):
- \( 10^{\frac{1}{4}} \) is the fourth root of 10, which is approximately \( 1.778 \).
- \( 10^{\frac{1}{3}} \) is the cube root of 10, which is approximately \( 2.154 \).

### Step 4: Compare with 2
Now we compare these values with 2:
1. Since \( 1.778 < 2 \), we have:
\[ 2 > 10^{\frac{1}{4}} \]
This confirms the first part of our inequality.

2. Since \( 2 < 2.154 \), we have:
\[ 2 < 10^{\frac{1}{3}} \]
This confirms the second part of our inequality.

### Conclusion
Since both inequalities hold true, we conclude that:
\[ \frac{1}{4} < \log_{10} 2 < \frac{1}{3} \]
Thus, we have proved that \( \log_{10} 2 \) lies between \( \frac{1}{4} \) and \( \frac{1}{3} \).

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