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.

( e^{2x} - 12e^x + 11 = 0 )

the solution set expressed in terms of logarithms is (\boxed{quad}).

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.

Explanation:

Step1: Isolate the exponential term

Given the equation \( e^{2x}-12e^{x}+11 = 0 \). Let \( y = e^{x} \), then the equation becomes a quadratic equation: \( y^{2}-12y + 11=0 \).

Step2: Solve the quadratic equation

Factor the quadratic equation: \( (y - 1)(y - 11)=0 \). So, \( y = 1 \) or \( y = 11 \).

Step3: Substitute back \( y = e^{x} \) and solve for \( x \)

• For \( y = 1 \): \( e^{x}=1 \). Taking the natural logarithm of both sides, we get \( \ln(e^{x})=\ln(1) \), which simplifies to \( x=\ln(1) = 0 \).
• For \( y = 11 \): \( e^{x}=11 \). Taking the natural logarithm of both sides, we get \( \ln(e^{x})=\ln(11) \), which simplifies to \( x = \ln(11) \approx 2.3979 \).

Answer:

The solution set in terms of natural logarithms is \( \{0, \ln(11)\} \), and the decimal approximations are \( 0, 2.3979 \) (rounded to four decimal places).