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-4*3^(6w)=-1750 What is the solution of the equation? Round your answer, if necessary, to the nearest thousandth. w~~

−4⋅36w=−1750 -4 \cdot 3^{6 w}=-1750 \newlineWhat is the solution of the equation?\newlineRound your answer, if necessary, to the nearest thousandth.\newlinew≈ w \approx

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Full solution

Q. −4⋅36w=−1750 -4 \cdot 3^{6 w}=-1750 \newlineWhat is the solution of the equation?\newlineRound your answer, if necessary, to the nearest thousandth.\newlinew≈ w \approx
  1. Isolate exponential term: First, we need to isolate the exponential term by dividing both sides of the equation by -4＂).\(\newline\$-4\cdot3^{(6w)} = -1750\)\(\newline\)\(3^{(6w)} = \frac{-1750}{-4}\)\(\newline\)\(3^{(6w)} = 437.5\)
  2. Apply logarithm: Next, we apply the logarithm to both sides of the equation to solve for the exponent. We can use the natural logarithm (ln) for this purpose.\(\newline\)\(\ln(3^{6w}) = \ln(437.5)\)
  3. Simplify using power property: Using the power property of logarithms, which states that \(\ln(a^b) = b\cdot\ln(a)\), we can simplify the left side of the equation.\(6w \cdot \ln(3) = \ln(437.5)\)
  4. Isolate w: Now, we isolate w by dividing both sides of the equation by \(6\ln(3)\). \(\newline\)\[w = \frac{\ln(437.5)}{6 \cdot \ln(3)}\]
  5. Perform calculations: We perform the calculations using a calculator to find the value of \(w\).\(w \approx \frac{\ln(437.5)}{(6 \times \ln(3))}\)\(w \approx \frac{2.681022}{(6 \times 1.098612)}\)\(w \approx \frac{2.681022}{6.591672}\)\(w \approx 0.406716\)
  6. Round the answer: Finally, we round the answer to the nearest thousandth as requested. \(w \approx 0.407\)

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