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Consider the equation 4*10^(-3x)=18 Solve the equation for x. Express the solution as a logarithm in base10. x= Approximate the value of x. Round your answer to the nearest thousandth. x~~

Consider the equation\newline4⋅10−3x=18 4 \cdot 10^{-3 x}=18 \newlineSolve the equation for x x . Express the solution as a logarithm in base1010.\newlinex= x= \newlineApproximate the value of x x . Round your answer to the nearest thousandth.\newlinex≈ x \approx

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Full solution

Q. Consider the equation\newline4⋅10−3x=18 4 \cdot 10^{-3 x}=18 \newlineSolve the equation for x x . Express the solution as a logarithm in base1010.\newlinex= x= \newlineApproximate the value of x x . Round your answer to the nearest thousandth.\newlinex≈ x \approx
  1. Isolate the exponential term: Isolate the exponential term.\newlineTo solve for xx, we first want to isolate the term with the exponent. We can do this by dividing both sides of the equation by 44.\newline4⋅10(−3x)=184 \cdot 10^{(-3x)} = 18\newline10(−3x)=18410^{(-3x)} = \frac{18}{4}\newline10(−3x)=4.510^{(-3x)} = 4.5
  2. Apply the logarithm: Apply the logarithm to both sides of the equation.\newlineTo solve for the exponent, we can take the logarithm of both sides of the equation. We will use the common logarithm (base 1010).\newlinelog⁡(10(−3x))=log⁡(4.5)\log(10^{(-3x)}) = \log(4.5)
  3. Use logarithm property: Use the property of logarithms to bring down the exponent.\newlineThe property of logarithms that we will use is log⁡(ba)=a⋅log⁡(b)\log(b^a) = a \cdot \log(b). This allows us to move the exponent in front of the logarithm.\newline−3x⋅log⁡(10)=log⁡(4.5)-3x \cdot \log(10) = \log(4.5)
  4. Simplify the equation: Simplify the left side of the equation.\newlineSince log⁡(10)\log(10) is equal to 11, the equation simplifies to:\newline−3x=log⁡(4.5)-3x = \log(4.5)
  5. Solve for x: Solve for x.\newlineTo solve for x, we divide both sides of the equation by −3-3.\newlinex=log⁡(4.5)−3x = \frac{\log(4.5)}{-3}
  6. Approximate the value of x: Approximate the value of x.\newlineUsing a calculator, we can find the value of log⁡(4.5)\log(4.5) and then divide by −3-3 to get the approximate value of x.\newlinex≈log⁡(4.5)−3x \approx \frac{\log(4.5)}{-3}\newlinex≈0.6532−3x \approx \frac{0.6532}{-3}\newlinex≈−0.2177x \approx -0.2177\newlineRounded to the nearest thousandth, x≈−0.218x \approx -0.218

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