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3*5^(0.2 w)=720 What is the solution of the equation? Round your answer, if necessary, to the nearest thousandth. w~~ ◻

3⋅50.2w=720 3 \cdot 5^{0.2 w}=720 \newlineWhat is the solution of the equation?\newlineRound your answer, if necessary, to the nearest thousandth.\newlinew≈ w \approx \newline□ \square

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Q. 3⋅50.2w=720 3 \cdot 5^{0.2 w}=720 \newlineWhat is the solution of the equation?\newlineRound your answer, if necessary, to the nearest thousandth.\newlinew≈ w \approx \newline□ \square
  1. Isolate variable term: First, let's isolate the term with the variable ww by dividing both sides of the equation by 33.5(0.2w)=72035^{(0.2w)} = \frac{720}{3}5(0.2w)=2405^{(0.2w)} = 240
  2. Get rid of exponent: Now, we need to get rid of the exponent. We can do this by taking the logarithm of both sides. Let's use the natural logarithm (ln) for this.\newlineln⁡(50.2w)=ln⁡(240)\ln(5^{0.2w}) = \ln(240)
  3. Take natural logarithm: Using the property of logarithms that allows us to move the exponent to the front, we get:\newline0.2w⋅ln⁡(5)=ln⁡(240)0.2w \cdot \ln(5) = \ln(240)
  4. Divide by ln⁡(5)\ln(5): Now, we'll divide both sides by ln⁡(5)\ln(5) to solve for ww.\newlinew=ln⁡(240)0.2×ln⁡(5)w = \frac{\ln(240)}{0.2 \times \ln(5)}
  5. Calculate final value: Let's do the calculation.\newlinew≈ln⁡(240)0.2×ln⁡(5)w \approx \frac{\ln(240)}{0.2 \times \ln(5)}\newlinew≈22.180709777918250.2×1.6094379124341003w \approx \frac{22.18070977791825}{0.2 \times 1.6094379124341003}\newlinew≈22.180709777918250.32188758248682006w \approx \frac{22.18070977791825}{0.32188758248682006}\newlinew≈68.906w \approx 68.906

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