The value of log 6 is equal to:

Options

A. log 1 + log 2 + log 3
B. log (1+2+3)
C. log (1 x 2 x 3)
D. all of the above

Answer

To find the value of log 6, we can express 6 in terms of its prime factors and use the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Express 6 in terms of its prime factors
We know that:
\[ 6 = 2 \times 3 \]

### Step 2: Apply the logarithm product rule
Using the property of logarithms that states:
\[ \log(ab) = \log a + \log b \]
we can write:
\[ \log 6 = \log(2 \times 3) = \log 2 + \log 3 \]

### Step 3: Recognize that log 1 is 0
We can also express 6 as:
\[ 6 = 1 + 2 + 3 \]
However, this doesn't directly help us find the logarithm. Instead, we can note that:
\[ \log 1 = 0 \]
Thus, we can also express:
\[ \log 6 = \log(1 \times 6) = \log 1 + \log 6 = 0 + \log 6 \]

### Step 4: Combine the results
From Step 2, we have:
\[ \log 6 = \log 2 + \log 3 \]
This shows that log 6 can be expressed in terms of the logarithms of its prime factors.

### Conclusion
Thus, we can conclude that log 6 can be expressed as:
\[ \log 6 = \log 2 + \log 3 \]
And since we can express it in multiple ways (as shown in the video), the answer is that log 6 can be represented in various forms.

### Final Answer
The value of log 6 can be expressed as:
- \( \log 6 = \log 2 + \log 3 \)
- \( \log 6 = \log(1 \times 6) = \log 1 + \log 6 \)

Hence, the correct option is **D: all of the above**.

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