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.
$2e^{2x} = 1282$

the solution set expressed in terms of logarithms is $left\\{ \frac{\ln(641)}{2} \
ight\\}$.
(use a comma to separate answers as needed. simplify your answer. use integers or fractions for any numbers in the equation. use ln for natural logarithm and log for common logarithm.)
now use a calculator to obtain a decimal approximation for the solution.
the solution set is $left\\{ \\ \
ight}$.
(use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Recall the expression

We have \( x = \frac{\ln(641)}{2} \).

Step2: Calculate the natural log

First, find \( \ln(641) \). Using a calculator, \( \ln(641) \approx 6.462 \).

Step3: Divide by 2

Then divide this value by 2: \( \frac{6.462}{2} \approx 3.23 \).

Answer:

\(\{3.23\}\)