Question

solve the equation.
$2.01e^t = 31.1$
$t = \square$
(round to four decimal places as needed.)

Explanation:

Step1: Divide both sides by 2.01

To isolate the exponential term, we divide both sides of the equation \(2.01e^{t}=31.1\) by 2.01. This gives us \(e^{t}=\frac{31.1}{2.01}\). Calculating the right - hand side, \(\frac{31.1}{2.01}\approx15.472637\).

Step2: Take the natural logarithm of both sides

Since \(y = e^{x}\) and \(x=\ln(y)\) are inverse functions, we take the natural logarithm of both sides of the equation \(e^{t}\approx15.472637\). So, \(\ln(e^{t})=\ln(15.472637)\). By the property of logarithms \(\ln(e^{x}) = x\), we get \(t=\ln(15.472637)\).

Step3: Calculate the value of t

Using a calculator to find the natural logarithm of 15.472637, we have \(t\approx2.7490\) (rounded to four decimal places).

Answer:

\(t\approx2.7490\)