Answer
(a) $k=\ln2$
(b) $k=\frac{\ln2}{10}$
(c) $k=\ln a^{1000}$
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) $$e^{2k}=4$$
- Recall this inverse property: $e^{\ln x}=x$
Therefore, we can rewrite $4=e^{\ln4}$
That means $$e^{2k}=e^{\ln4}$$ $$2k=\ln4=\ln2^2$$
- Apply Power Rule: $$2k=2\ln 2$$ $$k=\ln2$$
(b) $$100e^{10k}=200$$ $$e^{10k}=\frac{200}{100}=2$$
- Recall this inverse property: $e^{\ln x}=x$
Therefore, we can rewrite $2=e^{\ln2}$
That means $$e^{10k}=e^{\ln2}$$ $$10k=\ln2$$ $$k=\frac{\ln2}{10}$$
(c) $$e^{\frac{k}{1000}}=a$$
- Recall this inverse property: $e^{\ln x}=x$
Therefore, we can rewrite $a=e^{\ln a}$
That means $$e^{\frac{k}{1000}}=e^{\ln a}$$ $$\frac{k}{1000}=\ln a$$ $$k=1000\ln a$$
- Apply Power Rule: $$k=\ln a^{1000}$$
