Question
Write the equation in logarithmic form. \( 3^{5}=243 \) \( \log _{[?]}[\quad]=[\quad] \)
Ask by Tucker Burns. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\log_{3}(243) = 5
\]
Solution
To convert the exponential equation \( 3^{5} = 243 \) into its logarithmic form, you can use the definition of a logarithm.
The general relationship between exponents and logarithms is:
\[ b^c = a \quad \text{is equivalent to} \quad \log_b(a) = c \]
Applying this to your equation:
- **Base (\(b\))**: 3
- **Exponent (\(c\))**: 5
- **Result (\(a\))**: 243
So, the logarithmic form of the equation \( 3^{5} = 243 \) is:
\[
\log_{3}(243) = 5
\]
**Final Answer:**
\[
\log_{3}(243) = 5
\]
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Simplify this solution Extra Insights
To convert the equation \( 3^{5}=243 \) into logarithmic form, we express it as \( \log_{3}(243) = 5 \). This means that "the logarithm base 3 of 243 is equal to 5." You can think of logarithms as the inversion of exponentiation – they tell you about the relationship between the base, the result, and the exponent! If you're looking to practice more with logarithmic conversions, try experimenting with different bases and results. For instance, try converting equations like \( 2^{4} = 16 \) into logarithmic form. The more you play around with logarithms, the more comfortable you'll become with recognizing and converting these equations!
