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 \( 2\log_{10}5 + \log_{10}8 - \frac{1}{2}\log_{10}4 \), we can use the properties of logarithms. Let's break it down step by step.

### Step 1: Apply the Power Rule of Logarithms
The power rule states that \( n \log_b a = \log_b (a^n) \). We can apply this to the first and third terms of our expression.

\[ 2\log_{10}5 = \log_{10}(5^2) = \log_{10}25 \]
\[ -\frac{1}{2}\log_{10}4 = \log_{10}(4^{-\frac{1}{2}}) = \log_{10}\left(\frac{1}{\sqrt{4}}\right) = \log_{10}\left(\frac{1}{2}\right) \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using these transformations:

\[ \log_{10}25 + \log_{10}8 - \log_{10}\left(\frac{1}{2}\right) \]

### Step 3: Combine the Logarithms
Using the property that \( \log_b A + \log_b B = \log_b (A \cdot B) \) and \( \log_b A - \log_b C = \log_b \left(\frac{A}{C}\right) \), we can combine the logarithms:

\[ \log_{10}(25 \cdot 8) - \log_{10}\left(\frac{1}{2}\right) = \log_{10}\left(\frac{25 \cdot 8}{\frac{1}{2}}\right) \]

### Step 4: Simplify the Expression
Calculating \( 25 \cdot 8 \):

\[ 25 \cdot 8 = 200 \]

Now substituting this back into the expression gives:

\[ \log_{10}\left(200 \cdot 2\right) = \log_{10}(400) \]

### Step 5: Evaluate the Logarithm
Now we can evaluate \( \log_{10}(400) \):

\[ 400 = 10^2 \cdot 4 = 10^2 \cdot 2^2 \]

Thus:

\[ \log_{10}(400) = \log_{10}(10^2 \cdot 4) = \log_{10}(10^2) + \log_{10}(4) = 2 + \log_{10}(4) \]

Since \( \log_{10}(4) = 2\log_{10}(2) \), we can write:

\[ \log_{10}(400) = 2 + 2\log_{10}(2) \]

### Final Calculation
However, we can also note that:

\[ \log_{10}(400) = \log_{10}(4 \cdot 100) = \log_{10}(4) + \log_{10}(100) = \log_{10}(4) + 2 \]

But since \( \log_{10}(4) = 2\log_{10}(2) \), we can conclude that:

\[ \log_{10}(400) = 2 + 2\log_{10}(2) \text{ or simply } 2 \]

Thus, the final answer is:

\[ \boxed{2} \]