Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use \[ e^{2 x}-6 e^{x}+5=0 \] The solution set expressed in terms of logarithms is \( \square \) ]. Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression.
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Beyond the Answer
To solve the equation \( e^{2x} - 6e^{x} + 5 = 0 \), let’s make a substitution. Let \( y = e^{x} \). This transforms the equation into a standard quadratic form: \[ y^2 - 6y + 5 = 0 \] Now we can factor this equation: \[ (y - 1)(y - 5) = 0 \] Setting each factor to zero gives us: \[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \] \[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \] Now we revert to our original variable using \( y = e^{x} \): 1. From \( e^{x} = 1 \): \[ x = \ln(1) = 0 \] 2. From \( e^{x} = 5 \): \[ x = \ln(5) \] Thus, the solution set expressed in terms of logarithms is: \[ \{ 0, \ln(5) \} \]
