Solve the following equation using the fact that a^x = a^y implies x = y

Question

Solve the following equation using the fact that a^x = a^y implies x = y

               2

  -9x      x

e     = e     + 20

 

x = ?

 

(Use a comma to separate answers as needed. simplify your answer. type an integer or a fraction.)

 

 

Answer 1

e^(-9x) = e^(x^2) + 20

I don’t know how to solve this exactly.

However, a numerical approximate solution is easily done on GeoGebra (or a graphing calculator).

Graph f(x) = e^(x^2) + 20 and g(x) = e^(-9x) and find their intersection point, A.

Or, multiply both sides by e^(9x):

0 = e^(x^2 + 9x) + 20e^(9x) - 1

and graph h(x) = e^(x^2 + 9x) + 20e^(9x) - 1 and find it’s zero.

Here’s my GeoGebra sketch:
http://www.wyzant.com/resources/files/262819/hard_exponentials_equation

And the answer is: x ≈ -0.338922463539581

check:

e^(-9(-0.338922463539581)) =? e^((-0.338922463539581)^2) + 20

e^(3.05030217185623) =? e^(0.11486843629174) + 20

21.12172584945494 ≈ 1.121725849451126 + 20 √

Answer 2

Take ln on both sides

 

ln e^-9x  =  lne^{x^2} + ln20

 

-9x = x² + 3

 

x²  + 9x +3 = 0

 

Solve this quadratic equation

 

x = (-9 + √(81-12))2

 

x= -9/2 + (√69)2

:)