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Consider the equation 5*10^((z)/(4))=32". " Solve the equation for z. Express the solution as a logarithm in base10. z= Approximate the value of z. Round your answer to the nearest thousandth. z~~

Consider the equation\newline5⋅10z4=32.  5 \cdot 10^{\frac{z}{4}}=32 \text {. } \newlineSolve the equation for z z . Express the solution as a logarithm in base1010.\newlinez= z= \newlineApproximate the value of z z . Round your answer to the nearest thousandth.\newlinez≈ z \approx

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Q. Consider the equation\newline5⋅10z4=32.  5 \cdot 10^{\frac{z}{4}}=32 \text {. } \newlineSolve the equation for z z . Express the solution as a logarithm in base1010.\newlinez= z= \newlineApproximate the value of z z . Round your answer to the nearest thousandth.\newlinez≈ z \approx
  1. Write and Identify Equation: Write down the given equation and identify the goal.\newlineThe given equation is 5⋅10z4=325\cdot10^{\frac{z}{4}} = 32. We need to solve for zz.
  2. Isolate Exponential Term: Isolate the exponential term.\newlineTo isolate the exponential term, divide both sides of the equation by 55.\newline10z/4=32510^{z/4} = \frac{32}{5}
  3. Convert to Logarithmic Form: Convert the equation to logarithmic form.\newlineTo solve for zz, we can take the logarithm of both sides of the equation. We will use the base 1010 logarithm since our exponential base is 1010.\newlinelog⁡(10z/4)=log⁡(325)\log(10^{z/4}) = \log(\frac{32}{5})
  4. Apply Power Rule of Logarithms: Apply the power rule of logarithms.\newlineThe power rule of logarithms states that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b\cdot\log(a). We apply this rule to the left side of the equation.\newlinez4⋅log⁡(10)=log⁡(325)\frac{z}{4}\cdot\log(10) = \log\left(\frac{32}{5}\right)
  5. Simplify Left Side of Equation: Simplify the left side of the equation.\newlineSince log⁡(10)\log(10) is equal to 11, the equation simplifies to:\newlinez4=log⁡(325)\frac{z}{4} = \log\left(\frac{32}{5}\right)
  6. Solve for z: Solve for z.\newlineTo solve for z, multiply both sides of the equation by 44.\newlinez=4log⁡(325)z = 4\log\left(\frac{32}{5}\right)
  7. Calculate Approximate Value of z: Calculate the approximate value of z.\newlineUsing a calculator, find the value of log⁡(325)\log(\frac{32}{5}) and then multiply by 44 to get zz.\newlinez≈4⋅log⁡(325)z \approx 4\cdot\log(\frac{32}{5})\newlinez≈4⋅0.8062z \approx 4\cdot0.8062 (rounded to four decimal places)\newlinez≈3.2248z \approx 3.2248

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