Answer
(a) $k=\frac{1}{5}\ln\frac{1}{4}$
(b) $k=\ln\frac{1}{80}$
(c) $k=1$
Work Step by Step
To solve for $k$ in this exercise, keep in mind this property:
- If $e^x = e^a$ then $x=a$
(a) $$e^{5k}=\frac{1}{4}$$
- Recall this inverse property: $e^{\ln x}=x$
Therefore, we can rewrite $\frac{1}{4}=e^{\ln\frac{1}{4}}$
That means $$e^{5k}=e^{\ln\frac{1}{4}}$$ $$5k=\ln\frac{1}{4}$$ $$k=\frac{1}{5}\ln\frac{1}{4}$$
(b) $$80e^{k}=1$$ $$e^{k}=\frac{1}{80}$$
- Recall this inverse property: $e^{\ln x}=x$
Therefore, we can rewrite $\frac{1}{80}=e^{\ln\frac{1}{80}}$
That means $$e^{k}=e^{\ln\frac{1}{80}}$$ $$k=\ln\frac{1}{80}$$
(c) $$e^{(\ln0.8)k}=0.8$$
- Recall this inverse property: $e^{\ln x}=x$
Therefore, we can rewrite $0.8=e^{\ln 0.8}$
That means $$e^{(\ln0.8)k}=e^{\ln 0.8}$$ $$(\ln0.8)k=\ln 0.8$$ $$k=1$$
