Question
The sum \( \log _{5}(3)+\log _{5}(8) \) can also be expressed as a.) \( \log _{8}(15) \) b.) \( \log _{5}(24) \) c.) \( \log _{5}(11) \) d.) \( \log _{3}(40) \)
Ask by Malone Bond. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The sum \( \log_{5}(3) + \log_{5}(8) \) is equal to \( \log_{5}(24) \).
Solution
To simplify the expression \( \log_{5}(3) + \log_{5}(8) \), we can use the logarithmic property that states:
\[
\log_{b}(m) + \log_{b}(n) = \log_{b}(m \times n)
\]
Applying this property:
\[
\log_{5}(3) + \log_{5}(8) = \log_{5}(3 \times 8) = \log_{5}(24)
\]
Among the given options, \( \log_{5}(24) \) corresponds to option **b**.
**Answer:** b.) \( \log _{5}(24) \)
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Simplify this solution Extra Insights
To solve \( \log _{5}(3) + \log _{5}(8) \), we can use the property of logarithms that states \( \log_a(b) + \log_a(c) = \log_a(b \cdot c) \). Thus, we have: \[ \log_{5}(3) + \log_{5}(8) = \log_{5}(3 \cdot 8) = \log_{5}(24) \] So, the correct expression is \( \log _{5}(24) \), which corresponds to option b. Let's check: - The product \( 3 \cdot 8 = 24 \), confirming our calculation is correct. So, the answer is b.) \( \log _{5}(24) \).
