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
2^x - 14e^x + 13 = 0
the solution set expressed in terms of logarithms is {0, ln 13}
(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.)
now use a calculator to obtain a decimal approximation for the solution.
the solution set is
(use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Analyze the equation

The given equation is \( e^{2x}-14e^{x}+13 = 0 \). Let \( y = e^{x} \), then the equation becomes a quadratic equation: \( y^{2}-14y + 13=0 \).

Step2: Solve the quadratic equation

Factor the quadratic equation: \( (y - 1)(y - 13)=0 \). So, \( y = 1 \) or \( y = 13 \).

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

• For \( y = 1 \): \( e^{x}=1 \). Since \( e^{0}=1 \), we get \( x = 0 \).
• For \( y = 13 \): \( e^{x}=13 \). Take the natural logarithm of both sides: \( x=\ln(13) \).

Step4: Approximate \( \ln(13) \)

Using a calculator, \( \ln(13)\approx2.56 \) (rounded to two decimal places).

Answer:

The solution set in decimal approximation is \( \{0, 2.56\} \)