Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = \frac{\ln 6}{\ln 3} \)
Solution
To solve the equation \( 3^{x} = 6 \) for \( x \), you can use logarithms. Here's how you can do it step-by-step:
1. **Take the natural logarithm of both sides**:
\[
\ln(3^{x}) = \ln(6)
\]
2. **Use the logarithm power rule** (\( \ln(a^{b}) = b \ln(a) \)):
\[
x \cdot \ln(3) = \ln(6)
\]
3. **Solve for \( x \)**:
\[
x = \frac{\ln(6)}{\ln(3)}
\]
So, the solution is:
\[
x = \frac{\ln 6}{\ln 3}
\]
**Alternatively**, using logarithms with a different base (such as base 10), the solution can also be written as:
\[
x = \frac{\log 6}{\log 3}
\]
**Final Answer**:
\[
x = \frac{\ln\,6}{\,\ln\,3}
\]
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Simplify this solution Extra Insights
To solve the equation \( 3^{x}=6 \), we can take the logarithm of both sides. So, applying the logarithm, we get \( x \cdot \log(3) = \log(6) \). Now, isolating \( x \) gives us \( x = \frac{\log(6)}{\log(3)} \). Using a calculator, you'll find that this roughly equals \( x \approx 1.6309 \). Fun fact: This equation illustrates the concept of exponential growth! Exponential equations are all around us, from the way our savings can grow in a bank to how populations expand over time. Understanding how to manipulate these equations can help you in various fields such as finance and biology!
