Question
Use properties of logarithms to find the exact value of the expression. Do not use a calculator. \[ \log _{16} 2+\log _{16} 8 \]
Ask by Mejia Tucker. in the United States
Jan 22,2025
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Tutor-Verified Answer
Answer
The exact value of the expression is \(1\).
Solution
To find the exact value of the expression \(\log_{16} 2 + \log_{16} 8\) using properties of logarithms, follow these steps:
1. **Use the Product Property of Logarithms**:
The product property states that \(\log_b (M) + \log_b (N) = \log_b (MN)\). Applying this property:
\[
\log_{16} 2 + \log_{16} 8 = \log_{16} (2 \times 8) = \log_{16} 16
\]
2. **Evaluate the Logarithm**:
Since \(16\) is the base of the logarithm, \(\log_{16} 16\) asks, "To what power must \(16\) be raised to get \(16\)?" The answer is \(1\):
\[
\log_{16} 16 = 1
\]
Therefore, the exact value of the expression is:
**Answer:** \(1\)
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To find the exact value of \(\log_{16} 2 + \log_{16} 8\), we can use the property of logarithms that states \(\log_b a + \log_b c = \log_b (a \cdot c)\). First, we combine the two logarithms: \[ \log_{16} 2 + \log_{16} 8 = \log_{16} (2 \cdot 8) = \log_{16} 16. \] Next, we can simplify \(\log_{16} 16\). Since \(16\) is the base of the logarithm, we know that: \[ \log_{16} 16 = 1. \] Thus, the exact value of the expression \(\log_{16} 2 + \log_{16} 8\) is \[ \boxed{1}. \]
