Pregunta
Question 5 Evaluate the expression: \( \log _{1 / 2} 64 \).
Ask by Frank Weaver. in the United States
Mar 19,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( \log_{1/2} 64 = -6 \)
Solución
1. We start by letting the logarithmic expression equal \( x \):
\[
\log_{1/2} 64 = x
\]
2. By the definition of logarithms, this means:
\[
\left(\frac{1}{2}\right)^x = 64
\]
3. We can rewrite \( \frac{1}{2} \) as \( 2^{-1} \):
\[
\left(2^{-1}\right)^x = 64 \quad \Longrightarrow \quad 2^{-x} = 64
\]
4. Express \( 64 \) as a power of \( 2 \), noting that \( 64 = 2^6 \):
\[
2^{-x} = 2^6
\]
5. Since the bases are the same, we equate the exponents:
\[
-x = 6
\]
6. Solve for \( x \):
\[
x = -6
\]
Thus, the value of \( \log_{1/2} 64 \) is \( -6 \).
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Simplifique esta solución Beyond the Answer
To evaluate \( \log_{1/2} 64 \), we're essentially asking: "To what power must I raise \( 1/2 \) to get \( 64 \)?" Since \( 64 \) can be expressed as \( 2^6 \), we can rewrite \( 1/2 \) as \( 2^{-1} \). Therefore, the expression becomes \( \log_{2^{-1}} 2^6 \). Using the change of base formula, we have: \[ \log_{2^{-1}} 2^6 = \frac{\log_2 2^6}{\log_2 2^{-1}} = \frac{6}{-1} = -6. \] Thus, \( \log_{1/2} 64 = -6 \).
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