The value of `log(ab)^(2) - log(ac) + log(bc^(4)) - 3 log (bc)` is:

Options

A. 0
B. log b
C. log c
D. log a

Answer

To solve the expression \( \log(ab)^2 - \log(ac) + \log(bc^4) - 3\log(bc) \), we will use the properties of logarithms. Let's break it down step-by-step.

### Step 1: Apply the Power Rule
Using the property \( \log(a^b) = b \log(a) \), we can rewrite \( \log(ab)^2 \) as:
\[ \log(ab)^2 = 2\log(ab) \]

### Step 2: Expand \( \log(ab) \)
Using the property \( \log(a \cdot b) = \log(a) + \log(b) \), we can expand \( \log(ab) \):
\[ \log(ab) = \log(a) + \log(b) \]
Thus,
\[ 2\log(ab) = 2(\log(a) + \log(b)) = 2\log(a) + 2\log(b) \]

### Step 3: Rewrite the Expression
Now we can rewrite the original expression:
\[ 2\log(a) + 2\log(b) - \log(ac) + \log(bc^4) - 3\log(bc) \]

### Step 4: Expand \( \log(ac) \) and \( \log(bc^4) \)
For \( \log(ac) \):
\[ \log(ac) = \log(a) + \log(c) \]
For \( \log(bc^4) \):
\[ \log(bc^4) = \log(b) + \log(c^4) = \log(b) + 4\log(c) \]

### Step 5: Expand \( 3\log(bc) \)
For \( 3\log(bc) \):
\[ 3\log(bc) = 3(\log(b) + \log(c)) = 3\log(b) + 3\log(c) \]

### Step 6: Substitute Back into the Expression
Now substitute these expansions back into the expression:
\[ 2\log(a) + 2\log(b) - (\log(a) + \log(c)) + (\log(b) + 4\log(c)) - (3\log(b) + 3\log(c)) \]

### Step 7: Combine Like Terms
Now, let's combine the terms:
- For \( \log(a) \): \( 2\log(a) - \log(a) = \log(a) \)
- For \( \log(b) \): \( 2\log(b) + \log(b) - 3\log(b) = 0 \)
- For \( \log(c) \): \( -\log(c) + 4\log(c) - 3\log(c) = 0 \)

### Step 8: Final Result
After combining all the terms, we are left with:
\[ \log(a) + 0 + 0 = \log(a) \]

### Conclusion
The value of the expression \( \log(ab)^2 - \log(ac) + \log(bc^4) - 3\log(bc) \) simplifies to:
\[ \log(a) \]