Question
the exponetial equation 3^x=12 does not have an integer solutio round your solution to 2 decimal places
x= ____ times ____ _____
x= ____ times ____ _____
Answers
GPT-4o mini
To solve the exponential equation \(3^x = 12\), we can take the logarithm of both sides. Using the natural logarithm (ln) gives us:
\[
\ln(3^x) = \ln(12)
\]
By using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can rewrite the left side:
\[
x \cdot \ln(3) = \ln(12)
\]
Now, we can solve for \(x\):
\[
x = \frac{\ln(12)}{\ln(3)}
\]
Next, we can calculate this using logarithm values:
1. \(\ln(12) \approx 2.4849\)
2. \(\ln(3) \approx 1.0986\)
Now, we can divide these values:
\[
x \approx \frac{2.4849}{1.0986} \approx 2.26
\]
Rounded to two decimal places, we find:
\[
x \approx 2.26
\]
Thus, the answer to your question is:
\(x = 2.26\) times \(1\) (since it does not multiply by any other number in this specific context).
So, written in the requested format:
\(x = 2.26 \, \text{times} \, 1\)
\[
\ln(3^x) = \ln(12)
\]
By using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can rewrite the left side:
\[
x \cdot \ln(3) = \ln(12)
\]
Now, we can solve for \(x\):
\[
x = \frac{\ln(12)}{\ln(3)}
\]
Next, we can calculate this using logarithm values:
1. \(\ln(12) \approx 2.4849\)
2. \(\ln(3) \approx 1.0986\)
Now, we can divide these values:
\[
x \approx \frac{2.4849}{1.0986} \approx 2.26
\]
Rounded to two decimal places, we find:
\[
x \approx 2.26
\]
Thus, the answer to your question is:
\(x = 2.26\) times \(1\) (since it does not multiply by any other number in this specific context).
So, written in the requested format:
\(x = 2.26 \, \text{times} \, 1\)
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