Question

consider the equation $-5 \cdot e^{10t} = -30$. 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 \approx $

Explanation:

Step1: Isolate the exponential term

Divide both sides by -5.
$\frac{-5 \cdot e^{10t}}{-5} = \frac{-30}{-5}$
$e^{10t} = 6$

Step2: Convert to natural logarithm

Take $\ln$ of both sides.
$\ln(e^{10t}) = \ln(6)$
$10t = \ln(6)$

Step3: Solve for t

Divide both sides by 10.
$t = \frac{\ln(6)}{10}$

Step4: Approximate the value

Calculate the decimal value.
$\ln(6) \approx 1.791759$, so $t \approx \frac{1.791759}{10} = 0.1791759$

Answer:

$t = \frac{\ln(6)}{10}$
$t \approx 0.179$