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Consider the equation -5*e^(10 t)=-30. Solve the equation for t. Express the solution as a logarithm in base- e. t= ◻ Approximate the value of t. Round your answer to the nearest thousandth. t~~ ◻

Consider the equation −5⋅e10t=−30 -5 \cdot e^{10 t}=-30 .\newlineSolve the equation for t t . Express the solution as a logarithm in base- e e .\newlinet=□t = \square\newlineApproximate the value of t t . Round your answer to the nearest thousandth.\newlinet≈□t \approx \square\newline

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Q. Consider the equation −5⋅e10t=−30 -5 \cdot e^{10 t}=-30 .\newlineSolve the equation for t t . Express the solution as a logarithm in base- e e .\newlinet=□t = \square\newlineApproximate the value of t t . Round your answer to the nearest thousandth.\newlinet≈□t \approx \square\newline
  1. Isolate exponential term: Isolate the exponential term.\newlineStart with the equation −5e10t=−30-5e^{10t} = -30.\newlineDivide both sides by −5-5 to isolate the exponential term.\newlinee10t=−30−5e^{10t} = \frac{-30}{-5}\newlinee10t=6e^{10t} = 6
  2. Take natural logarithm: Take the natural logarithm of both sides.\newlineTo solve for tt, take the natural logarithm (ln⁡\ln) of both sides of the equation.\newlineln⁡(e10t)=ln⁡(6)\ln(e^{10t}) = \ln(6)
  3. Apply logarithm property: Apply the property of logarithms.\newlineUse the property of logarithms that ln⁡(ex)=x\ln(e^x) = x to simplify the left side of the equation.\newline10t=ln⁡(6)10t = \ln(6)
  4. Solve for t: Solve for t.\newlineDivide both sides by 1010 to solve for tt.\newlinet=ln⁡(6)10t = \frac{\ln(6)}{10}
  5. Approximate value of tt: Approximate the value of tt. Use a calculator to find the approximate value of ln⁡(6)\ln(6) and then divide by 1010. t≈ln⁡(6)/10t \approx \ln(6) / 10 t≈0.179/10t \approx 0.179/10 t≈0.018t \approx 0.018 However, rounding to the nearest thousandth, we get: t≈0.018t \approx 0.018

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