Expert: Sherry Wallin - 1/18/2010

Question


Find the value of k in the following equation:

[9.e^(-6.k) ] - [9.e^(6.k)] + [7.e^(12.k)] - 7 = 0

Please note that: The signs ^ is for the power of  e, . (dot) is for times,  - (minus) for negative or take away and e is not the alphabetical letter e. e is for common logarithm.


Answer
Hi Dag~
    This is really a 'factor  by grouping' kind of problem. I'll try to walk you through it step by step:

What do I need to multiply 9.e^(6.k) by to get -9.e^(-6.k)? Using laws of exponents you really want to know what to multiply -6k by to get 6k and that is 12k. So factor out a -9.e^(-6.k) from the first two terms getting:

-9.e^(-6.k)[-1 + e^(12.k)] which is equivalent to [e^(12.k) -1]and now factor out a 7 from the 2nd two terms getting:

7(e^(12.k) -1).

So now you have -9.e^(-6.k)[e^(12.k) -1] + 7(e^(12.k) -1) = 0. Hopefully you see you can factor out the [e^(12.k) -1][7-9.e^(-6.k)] = 0.

[e^(12.k) -1]= 0 -> e^(12.k) = 1 -> by taking the ln of both sides you get: 12k = 0 -> k = 0

[7-9.e^(-6.k)] = 0 -> -9.e^(-6.k)= -7 -> e^(-6.k) = 7/9 -> by taking the ln of both sides you get
-6k = ln(7/9) -> k = -ln(7/9)/6

Math Prof