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Consider the equation 0.3*e^(3x)=27". " Solve the equation for x. Express the solution as a logarithm in base e. x= Approximate the value of x. Round your answer to the nearest thousandth. x~~

Consider the equation\newline0.3⋅e3x=27.  0.3 \cdot e^{3 x}=27 \text {. } \newlineSolve the equation for x x . Express the solution as a logarithm in basee.\newlinex= x= \newlineApproximate the value of x x . Round your answer to the nearest thousandth.\newlinex≈ x \approx

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Q. Consider the equation\newline0.3⋅e3x=27.  0.3 \cdot e^{3 x}=27 \text {. } \newlineSolve the equation for x x . Express the solution as a logarithm in basee.\newlinex= x= \newlineApproximate the value of x x . Round your answer to the nearest thousandth.\newlinex≈ x \approx
  1. Isolate exponential term: Isolate the exponential term.\newlineTo solve for xx, we first need to isolate the exponential term e3xe^{3x}. We do this by dividing both sides of the equation by 0.30.3.\newlineCalculation: 0.3⋅e3x=27⇒e3x=270.3⇒e3x=900.3\cdot e^{3x} = 27 \Rightarrow e^{3x} = \frac{27}{0.3} \Rightarrow e^{3x} = 90
  2. Take natural logarithm: Take the natural logarithm of both sides.\newlineTo solve for the exponent, we take the natural logarithm (ln⁡\ln) of both sides of the equation because ln⁡\ln and ee are inverse functions.\newlineCalculation: ln⁡(e3x)=ln⁡(90)\ln(e^{3x}) = \ln(90)
  3. Apply logarithm property: Apply the property of logarithms.\newlineUsing the property that ln⁡(ey)=y\ln(e^y) = y, we can simplify the left side of the equation.\newlineCalculation: 3x=ln⁡(90)3x = \ln(90)
  4. Solve for x: Solve for x.\newlineNow, we divide both sides of the equation by 33 to solve for xx.\newlineCalculation: x=ln⁡(90)3x = \frac{\ln(90)}{3}
  5. Approximate x value: Approximate the value of xx. We use a calculator to find the numerical value of ln⁡(90)\ln(90) and then divide by 33 to get the value of xx. Calculation: x≈ln⁡(90)/3≈4.4998/3≈1.4999x \approx \ln(90) / 3 \approx 4.4998 / 3 \approx 1.4999 Rounded to the nearest thousandth: x≈1.500x \approx 1.500

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