Without using log tables evaluate: `2log_(10)5+log_(10)8-(1)/(2)log_(10)4`

Options

A. 1
B. 2
C. 5
D. 3

Explanation

To evaluate the expression \(2\log_{10}5 + \log_{10}8 - \frac{1}{2}\log_{10}4\) without using logarithm tables, we can apply the properties of logarithms step by step.

### Step-by-Step Solution:

1. **Write down the expression:**
\[ 2\log_{10}5 + \log_{10}8 - \frac{1}{2}\log_{10}4 \]

2. **Apply the power rule of logarithms:**
The power rule states that \(n \log_b a = \log_b (a^n)\). Therefore, we can rewrite \(2\log_{10}5\) and \(\frac{1}{2}\log_{10}4\):
\[ \log_{10}(5^2) + \log_{10}8 - \log_{10}(4^{1/2}) \]
This simplifies to:
\[ \log_{10}(25) + \log_{10}(8) - \log_{10}(2) \]

3. **Combine the logarithms using the property \( \log_a b + \log_a c = \log_a (bc) \):**
\[ \log_{10}(25 \cdot 8) - \log_{10}(2) \]

4. **Now apply the property \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \):**
\[ \log_{10}\left(\frac{25 \cdot 8}{2}\right) \]

5. **Calculate the expression inside the logarithm:**
\[ 25 \cdot 8 = 200 \quad \text{and} \quad \frac{200}{2} = 100 \]
Thus, we have:
\[ \log_{10}(100) \]

6. **Recognize that \(100\) can be expressed as \(10^2\):**
\[ \log_{10}(10^2) \]

7. **Apply the power rule again:**
\[ 2\log_{10}(10) \]

8. **Since \(\log_{10}(10) = 1\):**
\[ 2 \cdot 1 = 2 \]

### Final Answer:
\[ 2\log_{10}5 + \log_{10}8 - \frac{1}{2}\log_{10}4 = 2 \]