Solve the equation for the indicated unknown.

Question

Solve the equation for the indicated unknown. Write your answers exactly and then, where appropriate, approximate to four decimal places.

 

(e^n)^2-e^n=6 (Hint: try replacing e^n with another variable, e.g. u before solving)

 

Answer 1

(eⁿ)² - eⁿ = 6

 

Let u = eⁿ.  Then u² - u = 6

 

                         u² - u - 6 = 0

 

                         (u - 3)(u + 2) = 0

 

                         u = 3 or u = -2

 

eⁿ = 3 or eⁿ = -2

 

Since eⁿ is never negative, eⁿ = 3

 

                                      So, n = ln3 ≈ 1.0986

 

 

 

Answer 2

substitute x for e^n then solve the simple quadratic equation by factoring

 

x^2-x=6

x^2-x-6=0

(x-3)(x+2)=0

x=3 or -2

 

 

Sub back in e^n and then use properties of logarithms to solve.  Remember log base e is the same as natural log (ln).

e^n=3 or -2

ln3=n or n=ln(-2)