Question

solve using properties of logarithm round two decimal places1. 5e^ (-0.4t) = 1.506

Answer 1

To solve the exercise, we can use the following property of logarithms:

$\ln (e^x)=x$

Then, we can solve the equation like this:

$\begin{gathered} 1.5e^{-0.4t}=1.506 \\ \text{ Divide by 1.5 from both sides of the equation} \\ \frac{1.5e^{-0.4t}}{1.5}=\frac{1.506}{1.5} \\ e^{-0.4t}=1.004 \\ \text{ Apply }\ln \text{ from both sides of the equation} \\ \ln (e^{-0.4t})=\ln (1.004) \\ \text{ Apply the mentioned property of logarithms} \\ -0.4t=\ln (1.004) \\ \text{ Divide by -0.4 from both sides of the equation} \\ \frac{-0.4t}{-0.4}=\frac{\ln(1.004)}{-0.4} \\ t\approx-0.01\Rightarrow\approx\text{ it reads "approximately"} \end{gathered}$

Therefore, the solution of the equation rounded to two decimal places is -0.01.