# Solving $7 = 8 - 2e^{-3k}$

So, this is an assignment my friend and I have for our homework: $7 = 8 - 2e^{-3k}$

And the solution should be: $\frac{1}{3\ln(2)}$

But, I have no idea how they got there. I tried doing:

\begin{align*} 7 &= 8 - 2e^{-3k}\\ 1 &= 2e^{-3k}\\ \frac{1}{2} &= e^{-3k} \end{align*}

And now I don't know what to do next. Help!

PS: Neither of us was in school when they were learning this lesson. So, some description along the way will be greatly welcomed!

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From your last equation: remember that $\ln\,e^u=u$... Also, $\ln\frac1{s}=-\ln\,s$. –  J. M. is back. Sep 15 '11 at 19:14
Take reciprocals: $e^{3k} = 2$. Take logs: $3k = \ln 2$. Solve for $k$. –  Srivatsan Sep 15 '11 at 19:16
(You should probably review what a logarithm is and its basic properties.) –  anon Sep 15 '11 at 19:21
But this does not lead to the specified conclusion. Perhaps there was an error copying the assignment and/or solution? –  Henning Makholm Sep 15 '11 at 19:22
Yes, thought if you are sure this is the correct question, then you copied the answer wrong. –  Leonardo Fontoura Sep 15 '11 at 19:26

\begin{align} \frac12 & = e^{-3k} \\ \\ 2 & = e^{3k} \\ \\ \log_e 2 & = 3k \\ \\ \frac{\log_e 2}{3} & = k \end{align}
The logarithm of a positive real number $x$ with respect to the base $a \in (0,\infty)\setminus \{1\}$ is the unique real number $\log_a x$ with the property that $a^{\log_a x}=x$.
In your case you have $\frac{1}{2}=e^{-3k}$ which is equivalent to $e^{3k}=2$. $\ln$ is the notation for $\log_e$ so by the definition in the first paragraph $3k=\ln e^{3k}=\ln 2$.