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Solve for xx.\newline14−4x=9x−314^{-4x}=9^{x-3}\newlineRound your answer to the nearest thousandth. Do not round any intermediate computations.\newlinex=x=

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Q. Solve for xx.\newline14−4x=9x−314^{-4x}=9^{x-3}\newlineRound your answer to the nearest thousandth. Do not round any intermediate computations.\newlinex=x=
  1. Write Equation: Write down the equation.\newlineWe are given the equation 14−4x=9x−314^{-4x} = 9^{x-3}.
  2. Apply Logarithms: Apply logarithms to both sides of the equation.\newlineTaking the natural logarithm (ln⁡\ln) of both sides gives us:\newlineln⁡(14−4x)=ln⁡(9x−3)\ln(14^{-4x}) = \ln(9^{x-3})
  3. Use Power Property: Use the power property of logarithms.\newlineThe power property states that ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b \cdot \ln(a). We apply this property to both sides:\newline−4x⋅ln⁡(14)=(x−3)⋅ln⁡(9)-4x \cdot \ln(14) = (x-3) \cdot \ln(9)
  4. Distribute Logarithms: Distribute the logarithms on both sides.\newline−4x⋅ln⁡(14)=x⋅ln⁡(9)−3⋅ln⁡(9)-4x \cdot \ln(14) = x \cdot \ln(9) - 3 \cdot \ln(9)
  5. Rearrange to Isolate xx: Rearrange the terms to isolate xx. Bring all terms involving xx to one side and constants to the other side: −4x⋅ln⁡(14)−x⋅ln⁡(9)=−3⋅ln⁡(9)-4x \cdot \ln(14) - x \cdot \ln(9) = -3 \cdot \ln(9)
  6. Factor Out x: Factor out xx from the left side.x⋅(−4⋅ln⁡(14)−ln⁡(9))=−3⋅ln⁡(9)x \cdot (-4 \cdot \ln(14) - \ln(9)) = -3 \cdot \ln(9)
  7. Solve for x: Solve for x.\newlineDivide both sides by (−4⋅ln⁡(14)−ln⁡(9))(-4 \cdot \ln(14) - \ln(9)) to isolate x:\newlinex=−3⋅ln⁡(9)−4⋅ln⁡(14)−ln⁡(9)x = \frac{-3 \cdot \ln(9)}{-4 \cdot \ln(14) - \ln(9)}
  8. Calculate x Value: Calculate the value of x without rounding intermediate computations.\newlinex=−3⋅ln⁡(9)−4⋅ln⁡(14)−ln⁡(9)x = \frac{-3 \cdot \ln(9)}{-4 \cdot \ln(14) - \ln(9)}\newlinex≈−3⋅2.197224577−4⋅2.639057329−2.197224577x \approx \frac{-3 \cdot 2.197224577}{-4 \cdot 2.639057329 - 2.197224577}\newlinex≈−6.591673731−10.556229316−2.197224577x \approx \frac{-6.591673731}{-10.556229316 - 2.197224577}\newlinex≈−6.591673731−12.753453893x \approx \frac{-6.591673731}{-12.753453893}\newlinex≈0.516798668x \approx 0.516798668
  9. Round Final Answer: Round the final answer to the nearest thousandth. x≈0.517x \approx 0.517

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