Find x, if :
` 2 + log x = log 45 - log 2 + log 16 - 2 log 3. `

Answer

To solve the equation \( 2 + \log x = \log 45 - \log 2 + \log 16 - 2 \log 3 \), we will follow these steps:

### Step 1: Simplify the Right Side
We start with the right side of the equation:
\[ \log 45 - \log 2 + \log 16 - 2 \log 3 \]
Using the logarithmic property \( \log a - \log b = \log \left( \frac{a}{b} \right) \) and \( n \log a = \log(a^n) \), we can rewrite \( -2 \log 3 \) as \( \log(3^2) = \log 9 \).

Thus, we have:
\[ \log 45 - \log 2 + \log 16 - \log 9 \]

### Step 2: Combine the Logarithms
Now, we can combine the logarithms:
\[ \log \left( \frac{45 \times 16}{2 \times 9} \right) \]
This simplifies to:
\[ \log \left( \frac{720}{18} \right) = \log 40 \]

### Step 3: Rewrite the Left Side
Now, we rewrite the left side of the equation:
\[ 2 + \log x = \log 40 \]
We can express \( 2 \) as \( \log(10^2) = \log 100 \). Therefore, we can rewrite the left side as:
\[ \log 100 + \log x = \log 40 \]

### Step 4: Combine the Left Side
Using the property of logarithms \( \log a + \log b = \log(ab) \), we have:
\[ \log(100x) = \log 40 \]

### Step 5: Remove the Logarithm
Since the logarithms are equal, we can set the arguments equal to each other:
\[ 100x = 40 \]

### Step 6: Solve for x
Now, we solve for \( x \):
\[ x = \frac{40}{100} = 0.4 \]

Thus, the value of \( x \) is:
\[ \boxed{0.4} \]

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