Given `2 log_(10) x + 1 = log_(10) 250`, find :
(i) x
(ii) `log_(10) 2x`

Answer

To solve the equation \( 2 \log_{10} x + 1 = \log_{10} 250 \), we will follow these steps:

### Step 1: Rewrite the equation using logarithmic identities
We start with the equation:
\[ 2 \log_{10} x + 1 = \log_{10} 250 \]
We can use the property of logarithms that states \( a \log_b c = \log_b (c^a) \). Thus, we can rewrite \( 2 \log_{10} x \) as:
\[ \log_{10} (x^2) \]
Now, the equation becomes:
\[ \log_{10} (x^2) + 1 = \log_{10} 250 \]
Next, we can express \( 1 \) as \( \log_{10} 10 \):
\[ \log_{10} (x^2) + \log_{10} 10 = \log_{10} 250 \]

### Step 2: Combine the logarithms
Using the property that \( \log_b a + \log_b c = \log_b (a \cdot c) \), we can combine the left side:
\[ \log_{10} (x^2 \cdot 10) = \log_{10} 250 \]

### Step 3: Remove the logarithm
Since the logarithms are equal, we can set the arguments equal to each other:
\[ x^2 \cdot 10 = 250 \]

### Step 4: Solve for \( x^2 \)
Now, we can isolate \( x^2 \):
\[ x^2 = \frac{250}{10} = 25 \]

### Step 5: Solve for \( x \)
Taking the square root of both sides gives:
\[ x = \pm 5 \]
Since \( x \) must be positive (as logarithms of negative numbers are undefined), we have:
\[ x = 5 \]

### Step 6: Find \( \log_{10} (2x) \)
Now we need to find \( \log_{10} (2x) \):
\[ 2x = 2 \cdot 5 = 10 \]
Thus:
\[ \log_{10} (2x) = \log_{10} 10 \]
Using the property of logarithms, we know:
\[ \log_{10} 10 = 1 \]

### Final Answers:
(i) \( x = 5 \)
(ii) \( \log_{10} (2x) = 1 \)