Question
Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution.
13 Superscript x Baseline equals 71
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Part 1
The solution set expressed in terms of logarithms is StartSet nothing EndSet .
(Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use ln for natural logarithm and log for common logarithm.)
13 Superscript x Baseline equals 71
Question content area bottom
Part 1
The solution set expressed in terms of logarithms is StartSet nothing EndSet .
(Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use ln for natural logarithm and log for common logarithm.)
Answers
GPT-4o mini
To solve the exponential equation \( 13^x = 71 \), we can take the logarithm of both sides. We can use either natural logarithm (\(\ln\)) or common logarithm (\(\log\)). Here, I will use natural logarithm for this solution.
1. Take the natural logarithm of both sides:
\[
\ln(13^x) = \ln(71)
\]
2. Apply the power rule of logarithms, which states that \(\ln(a^b) = b \cdot \ln(a)\):
\[
x \cdot \ln(13) = \ln(71)
\]
3. Solve for \(x\):
\[
x = \frac{\ln(71)}{\ln(13)}
\]
Thus, the solution set expressed in terms of logarithms is:
\[
\left\{ \frac{\ln(71)}{\ln(13)} \right\}
\]
Now, let's calculate a decimal approximation for the solution using a calculator.
1. Calculate \(\ln(71)\) and \(\ln(13)\).
- \( \ln(71) \approx 4.2627 \)
- \( \ln(13) \approx 2.5649 \)
2. Compute \(x\):
\[
x \approx \frac{4.2627}{2.5649} \approx 1.664
\]
Therefore, the solution set is:
\[
\left\{ \frac{\ln(71)}{\ln(13)} \right\}
\]
And the decimal approximation for the solution is approximately \(1.664\).
1. Take the natural logarithm of both sides:
\[
\ln(13^x) = \ln(71)
\]
2. Apply the power rule of logarithms, which states that \(\ln(a^b) = b \cdot \ln(a)\):
\[
x \cdot \ln(13) = \ln(71)
\]
3. Solve for \(x\):
\[
x = \frac{\ln(71)}{\ln(13)}
\]
Thus, the solution set expressed in terms of logarithms is:
\[
\left\{ \frac{\ln(71)}{\ln(13)} \right\}
\]
Now, let's calculate a decimal approximation for the solution using a calculator.
1. Calculate \(\ln(71)\) and \(\ln(13)\).
- \( \ln(71) \approx 4.2627 \)
- \( \ln(13) \approx 2.5649 \)
2. Compute \(x\):
\[
x \approx \frac{4.2627}{2.5649} \approx 1.664
\]
Therefore, the solution set is:
\[
\left\{ \frac{\ln(71)}{\ln(13)} \right\}
\]
And the decimal approximation for the solution is approximately \(1.664\).