Question
\( \log 8+\log 125=\square \) (Simplify your answer.)
Ask by Zimmerman Estrada. in Saudi Arabia
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\log 8 + \log 125 = 3
\]
Solution
1. First, use the property of logarithms:
\[
\log a + \log b = \log (a \cdot b)
\]
2. Apply the property to the given expression:
\[
\log 8 + \log 125 = \log (8 \cdot 125)
\]
3. Calculate the product:
\[
8 \cdot 125 = 1000
\]
So,
\[
\log (8 \cdot 125) = \log 1000
\]
4. Recognize that \(1000 = 10^3\). Therefore:
\[
\log 1000 = \log (10^3)
\]
5. Use the logarithm power rule:
\[
\log (10^3) = 3 \log 10
\]
Given that \(\log 10 = 1\) (for base 10 logarithms), it follows that:
\[
3 \log 10 = 3
\]
Thus,
\[
\log 8+\log 125=3.
\]
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Simplify this solution Extra Insights
Did you know that you can combine logarithms using the product property? That's right! When you add logarithms with the same base, it’s like multiplying the numbers inside the logs. So, \( \log 8 + \log 125 = \log(8 \times 125) \). Now, let's do the math: \( 8 \times 125 = 1000 \). Therefore, \( \log 8 + \log 125 = \log 1000 \). If you're using base 10 (common logarithm), \( \log 1000 = 3 \), since \( 10^3 = 1000 \). So, the final answer is \( 3 \)!
