If a,b,c are three consecutive integers, then log(ac+1) has the value:

Options

A. log b
B. `(log b)^(2)`
C. 2 log b
D. log 2b

Answer

To solve the problem, we need to find the value of \( \log(ac + 1) \) given that \( a, b, c \) are three consecutive integers.

### Step-by-step Solution:

1. **Define the integers**:
Let the three consecutive integers be:
- \( a \) (the first integer)
- \( b = a + 1 \) (the second integer)
- \( c = a + 2 \) (the third integer)

2. **Calculate the product \( ac \)**:
We need to find \( ac \):
\[ ac = a(a + 2) = a^2 + 2a \]

3. **Add 1 to \( ac \)**:
Now, we will add 1 to \( ac \):
\[ ac + 1 = a^2 + 2a + 1 \]

4. **Factor the expression**:
Notice that \( a^2 + 2a + 1 \) can be factored:
\[ ac + 1 = (a + 1)^2 \]

5. **Substitute \( b \)**:
Since \( b = a + 1 \), we can substitute \( b \) into the expression:
\[ ac + 1 = b^2 \]

6. **Take the logarithm**:
Now we can find \( \log(ac + 1) \):
\[ \log(ac + 1) = \log(b^2) \]

7. **Use the logarithm power rule**:
Using the property of logarithms \( \log(x^n) = n \log(x) \):
\[ \log(b^2) = 2 \log(b) \]

### Final Result:
Thus, the value of \( \log(ac + 1) \) is:
\[ \log(ac + 1) = 2 \log(b) \]