Pregunta
Find the following without a calculator. \[ \log _{3} 81=[?] \]
Ask by Curry Rose. in the United States
Jan 24,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
\(\log_{3} 81 = 4\)
Solución
To find \(\log_{3} 81\) without a calculator, you're looking for the exponent \(x\) that satisfies the equation:
\[
3^{x} = 81
\]
Let's break it down:
1. **Express 81 as a Power of 3:**
We know that:
\[
3^{1} = 3
\]
\[
3^{2} = 9
\]
\[
3^{3} = 27
\]
\[
3^{4} = 81
\]
2. **Identify the Exponent:**
From the above, it's clear that:
\[
3^{4} = 81
\]
Therefore, the exponent \(x\) is 4.
3. **Conclusion:**
\[
\log_{3} 81 = 4
\]
**Answer:** \(4\)
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Bonus Knowledge
Did you know that \( 81 \) is actually \( 3^4 \)? That’s right! So when we set up our logarithm, we’re really looking at \( \log_3 (3^4) \). From the rules of logarithms, we know that \( \log_b (b^x) = x \), which means \( \log_3 (3^4) = 4 \). Voilà, \( \log_3 81 = 4 \)! Now, here’s a fun tip: whenever you're dealing with powers of the base in logarithms, look for the relationship. If you can express your number as a power of your base, it will make solving it a piece of cake! Remember, mastering the basics allows you to tackle even more complex problems down the road!
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