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Consider the equation 14*10^(0.5 w)=100. Solve the equation for w. Express the solution as a logarithm in base- 10 . w=◻ Approximate the value of w. Round your answer to the nearest thousandth. w~~◻

Consider the equation 14⋅100.5w=100 14 \cdot 10^{0.5 w}=100 .\newlineSolve the equation for w w . Express the solution as a logarithm in base- 1010 .\newlinew=□w = \square \newlineApproximate the value of w w . Round your answer to the nearest thousandth.\newlinew≈□w \approx \square

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Q. Consider the equation 14⋅100.5w=100 14 \cdot 10^{0.5 w}=100 .\newlineSolve the equation for w w . Express the solution as a logarithm in base- 1010 .\newlinew=□w = \square \newlineApproximate the value of w w . Round your answer to the nearest thousandth.\newlinew≈□w \approx \square
  1. Take Logarithm: Now, we will take the logarithm of both sides of the equation to remove the exponent on the left side. We will use the base−10-10 logarithm since we want to express the solution as a logarithm in base−10-10.\newlinelog⁡(100.5w)=log⁡(7.14285714…)\log(10^{0.5 w}) = \log(7.14285714\ldots)\newline0.5w⋅log⁡(10)=log⁡(7.14285714…)0.5 w \cdot \log(10) = \log(7.14285714\ldots)\newlineSince log⁡(10)\log(10) is 11, this simplifies to:\newline0.5w=log⁡(7.14285714…)0.5 w = \log(7.14285714\ldots)
  2. Divide and Solve: Next, we divide both sides of the equation by 0.50.5 to solve for ww.\newline0.5w/0.5=log⁡(7.14285714...)/0.50.5 w / 0.5 = \log(7.14285714...) / 0.5\newlinew=2×log⁡(7.14285714...)w = 2 \times \log(7.14285714...)
  3. Calculate Approximate Value: Now we can use a calculator to find the approximate value of ww. We will round the answer to the nearest thousandth.\newlinew≈2×log⁡(7.14285714...)w \approx 2 \times \log(7.14285714...)\newlinew≈2×0.853871964w \approx 2 \times 0.853871964\newlinew≈1.707743928w \approx 1.707743928\newlineRounded to the nearest thousandth:\newlinew≈1.708w \approx 1.708

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