3. Chapter 3 (Course Work Solve the logarithmic equation. Be sure to reject any value of \( x \) that is not in the domain of the \( \log (x+8)-\log 2=\log (2 x+3) \) Solve the equation to find the solution set. Select the correct choice below and, if necessary, \( f \) A. The solution set is \{.. \( \} \). (Simplify your answer. Use a comma to separate answers as needed.) B. There are infinitely many solutions. C. There is no solution.
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
Mind Expander
To solve the logarithmic equation \(\log(x + 8) - \log(2) = \log(2x + 3)\), we can use the properties of logarithms. First, we can combine the logs on the left side: \[ \log\left(\frac{x + 8}{2}\right) = \log(2x + 3) \] By using the property that if \(\log(a) = \log(b)\), then \(a = b\), we get: \[ \frac{x + 8}{2} = 2x + 3 \] Cross-multiply and simplify: \[ x + 8 = 4x + 6 \] Now, rearranging gives: \[ 8 - 6 = 4x - x \] \[ 2 = 3x \] \[ x = \frac{2}{3} \] Now we need to check if this solution is within the domain of the original logs. For \(\log(x + 8)\) to be defined, \(x + 8 > 0\), which is true for \(x = \frac{2}{3}\). For \(\log(2x + 3)\) to be defined, \(2x + 3 > 0\), which is also true for \(x = \frac{2}{3}\). Thus, the solution set is \(\{ \frac{2}{3} \}\). So the correct choice is: A. The solution set is \{ \( \frac{2}{3} \} \}.
