Question
Simplify \( \log_3 81 \) by expressing it using a change of base to base 10.
Ask by Johnston Clark. in Turkey
Nov 25,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \log_3 81 = \frac{\log_{10} 81}{\log_{10} 3} \)
Solution
To simplify \( \log_3 81 \) using a change of base to base 10, we can use the change of base formula:
\[ \log_a b = \frac{\log_c b}{\log_c a} \]
In this case, we want to change the base from 3 to 10. So, we have:
\[ \log_3 81 = \frac{\log_{10} 81}{\log_{10} 3} \]
Now, we can calculate the values of \( \log_{10} 81 \) and \( \log_{10} 3 \) to simplify the expression.
Calculate the value by following steps:
- step0: Calculate:
\(\log_{10}{\left(81\right)}\)
- step1: Write in exponential form:
\(\log_{10}{\left(3^{4}\right)}\)
- step2: Calculate:
\(4\log_{10}{\left(3\right)}\)
Calculate or simplify the expression \( \log_{10} 3 \).
Calculate the value by following steps:
- step0: Calculate:
\(\log_{10}{\left(3\right)}\)
The value of \( \log_{10} 81 \) is approximately 1.908485 and the value of \( \log_{10} 3 \) is \( \log_{10}{(3)} \).
Therefore, the simplified expression for \( \log_3 81 \) using a change of base to base 10 is:
\[ \log_3 81 = \frac{1.908485}{\log_{10}{(3)}} \]
This is the simplified form of \( \log_3 81 \) using a change of base to base 10.
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Simplify this solution The Deep Dive
To simplify \( \log_3 81 \) using a change of base to base 10, you can use the formula: \[ \log_b a = \frac{\log_k a}{\log_k b} \] In this case, \( a = 81 \), \( b = 3 \), and \( k = 10 \). So we have: \[ \log_3 81 = \frac{\log_{10} 81}{\log_{10} 3} \] Now, \( 81 \) can be expressed as \( 3^4 \), allowing us to compute \( \log_{10} 81 \): \[ \log_{10} 81 = \log_{10} (3^4) = 4 \cdot \log_{10} 3 \] Substituting this back into the change of base formula gives: \[ \log_3 81 = \frac{4 \cdot \log_{10} 3}{\log_{10} 3} = 4 \] Therefore, the simplified form of \( \log_3 81 \) is \( \boxed{4} \).
