Express as a single logarithm : `2 + (1)/(2) log_(10) 9 - 2 log_(10) 5`

Answer

To express the given expression \(2 + \frac{1}{2} \log_{10} 9 - 2 \log_{10} 5\) as a single logarithm, we can follow these steps:

### Step 1: Rewrite the constants and logarithmic terms
We start with the expression:
\[ f(x) = 2 + \frac{1}{2} \log_{10} 9 - 2 \log_{10} 5 \]
We can rewrite \(2\) as \(\log_{10} 100\) because \(100 = 10^2\):
\[ f(x) = \log_{10} 100 + \frac{1}{2} \log_{10} 9 - 2 \log_{10} 5 \]

### Step 2: Apply the power rule of logarithms
Using the property of logarithms that states \(a \log_b c = \log_b (c^a)\), we can rewrite \(\frac{1}{2} \log_{10} 9\) and \(2 \log_{10} 5\):
\[ \frac{1}{2} \log_{10} 9 = \log_{10} (9^{1/2}) = \log_{10} 3 \]
\[ 2 \log_{10} 5 = \log_{10} (5^2) = \log_{10} 25 \]
Now substituting these back into the expression gives:
\[ f(x) = \log_{10} 100 + \log_{10} 3 - \log_{10} 25 \]

### Step 3: Combine the logarithmic terms
Using the properties of logarithms, we can combine the logarithmic terms. The property \(\log_b A + \log_b B = \log_b (A \cdot B)\) allows us to combine the first two terms:
\[ f(x) = \log_{10} (100 \cdot 3) - \log_{10} 25 \]
Next, we can apply the subtraction property \(\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)\):
\[ f(x) = \log_{10} \left(\frac{100 \cdot 3}{25}\right) \]

### Step 4: Simplify the fraction
Now we simplify the expression inside the logarithm:
\[ \frac{100 \cdot 3}{25} = \frac{300}{25} = 12 \]
Thus, we have:
\[ f(x) = \log_{10} 12 \]

### Final Answer
The expression \(2 + \frac{1}{2} \log_{10} 9 - 2 \log_{10} 5\) can be expressed as a single logarithm:
\[ \log_{10} 12 \]