SOLUTION: Solve for x: {{{e^(3x+1)=1/(e^x-2)}}}

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve for x: {{{e^(3x+1)=1/(e^x-2)}}}      Log On


   

Question 130650: Solve for x: e%5E%283x%2B1%29=1%2F%28e%5Ex-2%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
e%5E%283x%2B1%29=1%2F%28e%5E%28x-2%29%29 Start with the given function


e%2Ae%5E%283x%29=1%2F%28e%5E%28x-2%29%29 Break up e%5E%283x%2B1%29 to get e%5E%283x%29%2Ae%5E1=e%2Ae%5E%283x%29


e%2Ae%5E%283x%29=1%2F%28e%5Ex%2Fe%5E2%29 Break up e%5E%28x-2%29 to get e%5E%28x%29%2Ae%5E%28-2%29=e%5Ex%2Fe%5E2

e%2Ae%5E%283x%29=e%5E2%2Fe%5Ex Flip the fraction on the right side


e%28e%5Ex%29%5E3=e%5E2%2Fe%5Ex Rewrite e%5E%283x%29 to get %28e%5Ex%29%5E3


Now let u=e%5Ex

e%2Au%5E3=e%5E2%2Fu Plug in u=e%5Ex


e%2Au%5E3%2Au=e%5E2 Multiply both sides by u


e%2Au%5E4=e%5E2 Multiply

u%5E4=e%5E2%2Fe Divide both sides by e

u%5E4=e Reduce


%28e%5Ex%29%5E4=e Now plug in u=e%5Ex


e%5E%284x%29=e Multiply


e%5E%284x%29=e%5E1 Rewrite e as e%5E1


Since the bases are equal, the exponents are equal

So 4x=1


x=1%2F4 Divide both sides by 4 to isolate x



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Answer:

So our solution is


x=1%2F4