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 21.61
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Part 1
Do you have to provide parenthesis if { } is already provided
The solution set expressed in terms of logarithms is
  
​(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 3.07 EndSet .
  

​(Use a comma to separate answers as needed. Round to two decimal places as​ needed.)

Answer 1

To solve the exponential equation \( e^x = 21.61 \), we can take the natural logarithm of both sides.

**Part 1:**

Taking the natural logarithm,

\[
x = \ln(21.61)
\]

So, the solution set expressed in terms of logarithms is:

\[
\{ \ln(21.61) \}
\]

**Part 2:**

Now, we will use a calculator to find the decimal approximation for \( x = \ln(21.61) \).

Calculating this gives:

\[
x \approx 3.07
\]

Therefore, the solution set in decimal form, rounded to two decimal places, is:

\[
\{ 3.07 \}
\]

In summary:
- The solution set in terms of logarithms is \( \{ \ln(21.61) \} \).
- The decimal approximation of the solution is \( \{ 3.07 \} \).