Pregunta
upstudy study bank question image url

Solve for \( x \). \[ 6^{-2 x}=17^{-x+6} \] Round your answer to the nearest thousandth. Do not round any intermediate computations.

Ask by Matthews Luna. in the United States
Mar 11,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

\( x \approx -22.656 \)

Solución

Solve the equation \( 6^{-2x}=17^{-x+6} \). Solve the equation by following steps: - step0: Solve for \(x\): \(6^{-2x}=17^{-x+6}\) - step1: Take the logarithm of both sides: \(\log_{6}{\left(6^{-2x}\right)}=\log_{6}{\left(17^{-x+6}\right)}\) - step2: Evaluate the logarithm: \(-2x=\left(-x+6\right)\times \log_{6}{\left(17\right)}\) - step3: Calculate: \(-2x=\log_{6}{\left(17\right)}\times \left(-x+6\right)\) - step4: Expand the expression: \(-2x=-\log_{6}{\left(17\right)}\times x+6\log_{6}{\left(17\right)}\) - step5: Move the variable to the left side: \(-2x+\log_{6}{\left(17\right)}\times x=6\log_{6}{\left(17\right)}\) - step6: Collect like terms: \(\left(-2+\log_{6}{\left(17\right)}\right)x=6\log_{6}{\left(17\right)}\) - step7: Change the signs: \(\left(2-\log_{6}{\left(17\right)}\right)x=-6\log_{6}{\left(17\right)}\) - step8: Divide both sides: \(\frac{\left(2-\log_{6}{\left(17\right)}\right)x}{2-\log_{6}{\left(17\right)}}=\frac{-6\log_{6}{\left(17\right)}}{2-\log_{6}{\left(17\right)}}\) - step9: Divide the numbers: \(x=-6\log_{\frac{36}{17}}{\left(17\right)}\) The solution to the equation \(6^{-2x}=17^{-x+6}\) is \(x=-6\log_{\frac{36}{17}}{(17)}\). To round the answer to the nearest thousandth, we need to calculate the value of \(x\) and then round it to the nearest thousandth. Let's calculate the value of \(x\): \[x=-6\log_{\frac{36}{17}}{(17)}\] Now, let's calculate the value of \(x\) and round it to the nearest thousandth. Calculate the value by following steps: - step0: Calculate: \(-6\log_{\frac{36}{17}}{\left(17\right)}\) The value of \(x\) is approximately \(-22.656475\). Rounding this value to the nearest thousandth, we get \(x \approx -22.656\).

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Beyond the Answer

To solve the equation \[ 6^{-2x} = 17^{-x+6}, \] we'll take the logarithm of both sides. Let's use the natural logarithm, denoted as \(\ln\): \[ \ln(6^{-2x}) = \ln(17^{-x+6}). \] Using the property of logarithms that \(\ln(a^b) = b \ln(a)\), we can rewrite both sides: \[ -2x \ln(6) = (-x + 6) \ln(17). \] Expanding the right side gives: \[ -2x \ln(6) = -x \ln(17) + 6 \ln(17). \] Now, we will rearrange the equation to isolate \(x\): \[ -2x \ln(6) + x \ln(17) = 6 \ln(17). \] Factoring \(x\) out from the left side: \[ x ( \ln(17) - 2 \ln(6) ) = 6 \ln(17). \] Now, solve for \(x\): \[ x = \frac{6 \ln(17)}{\ln(17) - 2 \ln(6)}. \] Next, we compute the values of the logarithms: \[ \ln(17) \approx 2.833213, \] \[ \ln(6) \approx 1.791759. \] Substituting these values into our equation, we find: \[ x = \frac{6 \cdot 2.833213}{2.833213 - 2 \cdot 1.791759}. \] Calculating the denominator: \[ 2.833213 - 2 \cdot 1.791759 = 2.833213 - 3.583518 \approx -0.750305. \] Now substituting this back into our equation for \(x\): \[ x = \frac{17.000278}{-0.750305} \approx -22.651. \] Rounding to the nearest thousandth, we find: \[ \boxed{-22.651}. \]

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad