Pregunta
Use the properties of logarithms to evaluate each of the following expressions. (a) \( \log _{6} 9+2 \log _{6} 2=\square \) (b) \( \ln e^{8}-\ln e^{2} \) (b)
Ask by Lang Donnelly. in the United States
Feb 27,2025
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(a) 2
(b) 6
Solución
We can use the properties of logarithms to combine and simplify the expressions.
(a) For log₆ 9 + 2 log₆ 2:
Step 1. Use the power rule on the second term:
2 log₆ 2 = log₆ (2²) = log₆ 4.
Step 2. Use the product rule on the combined terms:
log₆ 9 + log₆ 4 = log₆ (9·4) = log₆ 36.
Step 3. Recognize that 36 is 6²:
log₆ 36 = log₆ (6²) = 2.
Thus, the expression evaluates to 2.
(b) For ln e⁸ – ln e²:
Step 1. Use the quotient rule for logarithms:
ln e⁸ – ln e² = ln (e⁸/e²).
Step 2. Simplify the exponent:
e⁸/e² = e^(8–2) = e⁶.
Step 3. Use the fact that ln (e^x) = x:
ln e⁶ = 6.
Thus, the expression evaluates to 6.
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Simplifique esta solución Beyond the Answer
To evaluate \( \log_{6} 9 + 2 \log_{6} 2 \), we apply the properties of logarithms. First, recognize that \( 2 \log_{6} 2 \) can be rewritten as \( \log_{6} (2^2) = \log_{6} 4 \). Therefore, our expression becomes \( \log_{6} 9 + \log_{6} 4 = \log_{6} (9 \cdot 4) = \log_{6} 36 \). Since \( 36 = 6^2 \), we have \( \log_{6} 36 = 2 \). For \( \ln e^{8} - \ln e^{2} \), we can use the property that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). That gives us \( \ln \left( \frac{e^{8}}{e^{2}} \right) = \ln \left( e^{8-2} \right) = \ln e^{6} \). Since \( \ln e^{x} = x \), we finally find that \( \ln e^{6} = 6 \).
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