Simplify: `3log_(3)5 + log_(3)10 - log_(3)625`.

Answer

To simplify the expression \(3\log_{3}5 + \log_{3}10 - \log_{3}625\), we will follow these steps:

### Step 1: Rewrite the first term
Using the property of logarithms that states \(n \log_b m = \log_b (m^n)\), we can rewrite the first term:
\[ 3\log_{3}5 = \log_{3}(5^3) = \log_{3}(125) \]

### Step 2: Combine the logarithms
Now we can combine the logarithms using the property that states \(\log_b m + \log_b n = \log_b (mn)\):
\[ \log_{3}(125) + \log_{3}(10) = \log_{3}(125 \times 10) = \log_{3}(1250) \]

### Step 3: Subtract the logarithm
Next, we will subtract \(\log_{3}625\) from \(\log_{3}(1250)\) using the property \(\log_b m - \log_b n = \log_b \left(\frac{m}{n}\right)\):
\[ \log_{3}(1250) - \log_{3}(625) = \log_{3}\left(\frac{1250}{625}\right) \]

### Step 4: Simplify the fraction
Now we simplify \(\frac{1250}{625}\):
\[ \frac{1250}{625} = 2 \]

### Step 5: Final logarithm
Thus, we have:
\[ \log_{3}\left(\frac{1250}{625}\right) = \log_{3}(2) \]

### Final Answer
The simplified expression is:
\[ \log_{3}(2) \]

---