Question

e^2x+3e^x-10=0
Solve the exponential equation. Express the solution as an exact answer in terms of natural logarithms. Then use a calculator to give an approximation of the solution, rounded to two decimal places.

Answer 1

Well hello there Miss Britt c:
\[\Large\rm e^{2x}+3e^x-10=0\]

Answer 2

Recall one of your exponent rules:\[\Large\rm (a^b)^c=a^{bc}\]Example:\[\Large\rm (x^3)^2=x^{3\cdot2}=x^6\]

Answer 3

We want to use this in reverse:\[\Large\rm e^{2\cdot x}=(e^x)^2\]

Answer 4

So we have:\[\Large\rm (e^x)^2+3(e^x)-10=0\]

Answer 5

What happens if we let \(\Large\rm u=e^x\) or something like that?
We get,\[\Large\rm (u)^2+3(u)-10=0\]\[\Large\rm u^2+3u-10=0\]

Answer 6

Ooo so it's really just a quadratic in disguise!
Maybe it factors!
Does it? :O hmmm you tell me..

Answer 7

mhm

Answer 8

(u+5)(u-2) Sorry I fell asleep last night ha :(

Answer 9

\[\Large\rm \left(\color{orangered}{u}+5\right)\left(\color{orangered}{u}-2\right)=0\]Which leads to\[\Large\rm u=-5\]\[\Large\rm u=2\]Undoing our substitution,\[\Large\rm e^x=-5\]\[\Large\rm e^x=2\]Take the natural log of each side to solve for x.
One of these solutions should end up being extraneous.