Answer
$a)$ $-100\ln1000$
$b)$ $\frac{-1}{k}\ln10$
$c)$ $-1$
Work Step by Step
$a)$
$e^{-0.01t}$ = $1000$
$-0.01t$ = $\ln1000$
$t$ = $\frac{1}{-0.01}\ln1000$
$t$ = $-100\ln1000$
$b)$
$e^{kt}$ = $\frac{1}{10}$
$kt$ = $\ln(\frac{1}{10})$
$t$ = $\frac{1}{k}\ln(10^{-1})$
$t$ = $\frac{-1}{k}\ln10$
$c)$
$e^{(\ln2)t}$ = $\frac{1}{2}$
${(\ln2)t}$ = $\ln\frac{1}{2}$
${(\ln2)t}$ = $\ln(2^{-1})$
$t$ = $-\frac{\ln2}{\ln2}$
$t$ = $-1$
