Question
Solve the exponential equation. Express the solution in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution.
e Superscript x Baseline equals 16.01
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Part 1
The solution set expressed in terms of logarithms is StartSet ln 16.01 EndSet .
(Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any numbers in the expression. Use ln for natural logarithm and log for common logarithm.)
Part 2
Now use a calculator to obtain a decimal approximation for the solution.
The solution set is StartSet 2.7 EndSet .
(Use a comma to separate answers as needed. Round to two decimal places as needed.)
e Superscript x Baseline equals 16.01
Question content area bottom
Part 1
The solution set expressed in terms of logarithms is StartSet ln 16.01 EndSet .
(Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any numbers in the expression. Use ln for natural logarithm and log for common logarithm.)
Part 2
Now use a calculator to obtain a decimal approximation for the solution.
The solution set is StartSet 2.7 EndSet .
(Use a comma to separate answers as needed. Round to two decimal places as needed.)
Answers
GPT-4o mini
To solve the equation \( e^x = 16.01 \), we take the natural logarithm of both sides:
\[
\ln(e^x) = \ln(16.01)
\]
Using the property of logarithms that states \(\ln(e^x) = x\), we have:
\[
x = \ln(16.01)
\]
**Part 1: Solution in terms of logarithms**
The solution set expressed in terms of logarithms is:
\[
\{ \ln(16.01) \}
\]
**Part 2: Decimal approximation**
Now, we can calculate the decimal approximation of \( \ln(16.01) \) using a calculator:
\[
\ln(16.01) \approx 2.7725887
\]
Rounding to two decimal places, we get:
\[
\approx 2.77
\]
Thus, the solution set in decimal approximation is:
\[
\{ 2.77 \}
\]
Summarizing, Part 1 is \( \{ \ln(16.01) \} \) and Part 2 is \( \{ 2.77 \} \).
\[
\ln(e^x) = \ln(16.01)
\]
Using the property of logarithms that states \(\ln(e^x) = x\), we have:
\[
x = \ln(16.01)
\]
**Part 1: Solution in terms of logarithms**
The solution set expressed in terms of logarithms is:
\[
\{ \ln(16.01) \}
\]
**Part 2: Decimal approximation**
Now, we can calculate the decimal approximation of \( \ln(16.01) \) using a calculator:
\[
\ln(16.01) \approx 2.7725887
\]
Rounding to two decimal places, we get:
\[
\approx 2.77
\]
Thus, the solution set in decimal approximation is:
\[
\{ 2.77 \}
\]
Summarizing, Part 1 is \( \{ \ln(16.01) \} \) and Part 2 is \( \{ 2.77 \} \).