Question

complete parts a) and b) for the exponential equation $e^x = 3$.
a) write an exact simplified solution using a natural logarithm.
b) approximate the solution by rounding to four decimal places.

(round to four decimal places as needed.)
a) the exact solution to the equation $e^x = 3$ using a natural logarithm is $x = \square$.
(type an exact answer in simplified form.)
b) the approximate solution to the equation $e^x = 3$ rounded to four decimal places is $x \approx \square$.

Explanation:

Step1: Solve for exact solution

To solve the equation \( e^x = 3 \) for \( x \) using natural logarithm, we take the natural logarithm of both sides. Recall that \( \ln(e^x)=x \) (by the property of logarithms \( \ln(a^b)=b\ln(a) \), here \( a = e \), \( b = x \), and \( \ln(e)=1 \)). So, taking \( \ln \) of both sides:
\( \ln(e^x)=\ln(3) \)
Which simplifies to \( x = \ln(3) \).

Step2: Approximate the solution

Now, we need to find the approximate value of \( \ln(3) \) rounded to four decimal places. Using a calculator, \( \ln(3)\approx1.0986 \).

Answer:

a) \( \ln(3) \)
b) \( 1.0986 \)