`log_(25) 625 - log_(31) 961 + log_(29) 841 = ?`

Options

A. 2
B. 6
C. 0
D. 4

Answer

To solve the expression \( \log_{25} 625 - \log_{31} 961 + \log_{29} 841 \), we will use properties of logarithms.

### Step 1: Rewrite the logarithms using exponentiation
We can express the numbers in terms of their bases raised to powers:
- \( 625 = 25^2 \)
- \( 961 = 31^2 \)
- \( 841 = 29^2 \)

Thus, we can rewrite the expression as:
\[ \log_{25}(25^2) - \log_{31}(31^2) + \log_{29}(29^2) \]

### Step 2: Apply the power rule of logarithms
Using the power rule of logarithms, which states that \( \log_a(b^n) = n \cdot \log_a(b) \), we can simplify each term:
\[ \log_{25}(25^2) = 2 \cdot \log_{25}(25) \]
\[ \log_{31}(31^2) = 2 \cdot \log_{31}(31) \]
\[ \log_{29}(29^2) = 2 \cdot \log_{29}(29) \]

### Step 3: Substitute the simplified terms back into the expression
Now substituting back, we have:
\[ 2 \cdot \log_{25}(25) - 2 \cdot \log_{31}(31) + 2 \cdot \log_{29}(29) \]

### Step 4: Evaluate the logarithms
Since \( \log_a(a) = 1 \) for any base \( a \):
\[ \log_{25}(25) = 1, \quad \log_{31}(31) = 1, \quad \log_{29}(29) = 1 \]

Substituting these values gives:
\[ 2 \cdot 1 - 2 \cdot 1 + 2 \cdot 1 \]

### Step 5: Simplify the expression
Now simplifying:
\[ 2 - 2 + 2 = 2 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{2} \]