Question

solve the equation e^x+25e^-x/2=5
a) what is the solution in terms of natural logarithms
b)what is the decimal approximation for the solution

Answer 1

Is the whole left side being divided by 2, or only the second term?

Answer 2

Or is the 2 in the exponent?
It's not very clear, grr...

Answer 3

the whole left side is being divided by 2

Answer 4

\[\Large\rm \frac{e^x+25e^{-x}}{2}=5\]Hmm ok ok I think we can work with this :)

Answer 5

Multiply both sides by 2,\[\Large\rm e^x+25e^{-x}=10\]This is the tricky step I guess,
multiply each side by e^x,\[\Large\rm (e^x)^2+25=10e^x\]Understand those steps so far?

Answer 6

yes, I understand so far

Answer 7

If we subtract the 10e^x over to the left side, we actually have a quadratic!\[\Large\rm (e^x)^2-10e^x+25=0\]You can make a substitution from here if you like..
Might make it easier to look at.\[\Large\rm u=e^x\]Plugging that in gives us:\[\Large\rm u^2-10u+25=0\]

Answer 8

Looks like this one will factor nicely, yes?

Answer 9

yes, it does

Answer 10

So what do you get for your factors? c:
You'll have just a little bit to do after that.
Because we've only found factors for `u` at this point.

Answer 11

I apologize my head is a little frazzled from doing math for a while

Answer 12

lol :)

Answer 13

Successsss!
That is our goal in teaching and assigning math work, to frazzle.
Learning? Pshh that's an afterthought.

Mission accomplished, student successfully frazzled.
Ok ok, jk but seriously, how bout them factors?
25 is ummm -5 times -5 right?
And -10 is -5 + -5?
Those work out, yes?

Answer 14

lol, yes those work

Answer 15

\[\Large\rm (u-5)(u-5)=0\]Ok so we get one solution, repeated.\[\Large\rm u=5\]Undoing our substitution gives us:\[\Large\rm e^x=5\]From here, how do we solve for x?
Hmmmm, whadya think?

Answer 16

Comeon Tams! Get your head in the game!
We have an exponential... maybeeeee something involving logs? *cough*

Answer 17

so, would it be ln(e^(5)

Answer 18

No.
Taking the log of each side gives us:\[\Large\rm \ln e^x=\ln 5\]Left side simplifies:\[\Large\rm x=\ln 5\]I'm not sure how your e ended in the log with the 5.. hmm

Answer 19

I'm not sure either, but was just reworking it on my side and just came to the same solution

Answer 20

Ah ok :)
So that solves part A!
For part B just go to your handy calculator.

Yay team!

Answer 21

Thank you very much for your time and assistance

Answer 22

np c: