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Consider the equation 0.75*10^((w)/(3))=30". " Solve the equation for w. Express the solution as a logarithm in base10. w= Approximate the value of w. Round your answer to the nearest thousandth. w~~

Consider the equation\newline0.75⋅10w3=30.  0.75 \cdot 10^{\frac{w}{3}}=30 \text {. } \newlineSolve the equation for w w . Express the solution as a logarithm in base1010.\newlinew= w= \newlineApproximate the value of w w . Round your answer to the nearest thousandth.\newlinew≈ w \approx

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Full solution

Q. Consider the equation\newline0.75⋅10w3=30.  0.75 \cdot 10^{\frac{w}{3}}=30 \text {. } \newlineSolve the equation for w w . Express the solution as a logarithm in base1010.\newlinew= w= \newlineApproximate the value of w w . Round your answer to the nearest thousandth.\newlinew≈ w \approx
  1. Write Equation, Isolate Exponential Term: Write down the given equation and isolate the exponential term.\newlineGiven equation: 0.75×10w/3=300.75\times10^{w/3} = 30\newlineTo isolate the exponential term, divide both sides by 0.750.75:\newline10w/3=300.7510^{w/3} = \frac{30}{0.75}
  2. Simplify Right Side: Simplify the right side of the equation.\newline300.75=40\frac{30}{0.75} = 40\newlineSo, the equation becomes:\newline10w3=4010^{\frac{w}{3}} = 40
  3. Apply Logarithm, Solve for ww: Apply the logarithm to both sides of the equation to solve for ww. We use the base 1010 logarithm because our exponential base is 1010. log⁡(10w/3)=log⁡(40)\log(10^{w/3}) = \log(40)
  4. Multiply by 33: Use the property of logarithms that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b\cdot\log(a).
    w3⋅log⁡(10)=log⁡(40)\frac{w}{3} \cdot \log(10) = \log(40)
    Since log⁡(10)\log(10) is 11, the equation simplifies to:
    w3=log⁡(40)\frac{w}{3} = \log(40)
  5. Calculate ww: Multiply both sides by 33 to solve for ww.\newlinew=3×log⁡(40)w = 3 \times \log(40)
  6. Round to Nearest Thousandth: Calculate the value of ww using a calculator.w≈3×log⁡(40)w \approx 3 \times \log(40)w≈3×1.60206w \approx 3 \times 1.60206 (using a calculator for log⁡(40)\log(40))w≈4.80618w \approx 4.80618
  7. Round to Nearest Thousandth: Calculate the value of ww using a calculator.w≈3×log⁡(40)w \approx 3 \times \log(40)w≈3×1.60206w \approx 3 \times 1.60206 (using a calculator for log⁡(40)\log(40))w≈4.80618w \approx 4.80618Round the value of ww to the nearest thousandth.w≈4.806w \approx 4.806

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