The value of `(log_(2)8 + log_(3)9 + log_(5) 25)` is:

Options

A. 5
B. 6
C. 7
D. none of these

Answer

To solve the expression \( \log_{2}8 + \log_{3}9 + \log_{5}25 \), we will evaluate each logarithm step by step.

### Step 1: Evaluate \( \log_{2}8 \)

We know that \( 8 = 2^3 \). Using the property of logarithms that states \( \log_{a}(b^c) = c \cdot \log_{a}(b) \), we can write:

\[ \log_{2}8 = \log_{2}(2^3) = 3 \cdot \log_{2}(2) \]

Since \( \log_{2}(2) = 1 \), we have:

\[ \log_{2}8 = 3 \cdot 1 = 3 \]

### Step 2: Evaluate \( \log_{3}9 \)

Similarly, we know that \( 9 = 3^2 \). Therefore, we can express it as:

\[ \log_{3}9 = \log_{3}(3^2) = 2 \cdot \log_{3}(3) \]

Again, since \( \log_{3}(3) = 1 \), we find:

\[ \log_{3}9 = 2 \cdot 1 = 2 \]

### Step 3: Evaluate \( \log_{5}25 \)

Next, we know that \( 25 = 5^2 \). Thus, we can write:

\[ \log_{5}25 = \log_{5}(5^2) = 2 \cdot \log_{5}(5) \]

Since \( \log_{5}(5) = 1 \), we have:

\[ \log_{5}25 = 2 \cdot 1 = 2 \]

### Step 4: Combine the results

Now, we can add all the evaluated logarithms together:

\[ \log_{2}8 + \log_{3}9 + \log_{5}25 = 3 + 2 + 2 \]

Calculating this gives:

\[ 3 + 2 + 2 = 7 \]

### Final Answer

Thus, the value of \( \log_{2}8 + \log_{3}9 + \log_{5}25 \) is \( \boxed{7} \).

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