4e^(2x) +e^(x) =14

Question

Solve the exponential equation. Express solution in exact form.

Answer 1

Subtract 14 on both sides of the equation.

 

4e^2x + e^x - 14 = 0

 

4(e^x)² + e^x - 14 = 0

 

 

Let   q = e^x .

 

4q² + q - 14 = 0

 

(4q - 7)(q + 2) = 0

 

 

Solving for q,

 

q = 7/4         ,          q = -2

 

 

Then we substitute the positive value of q into the original substitution.

 

7/4 = e^x

 

 

log both sides of the equation and bring any exponents down as the coefficient of the log.

 

ln(7/4) = xln(e)

 

ln(7/4) = x

Answer 2

a=e^x a quadratic in disguise

4a² + a -14=0

Factor 

(4a-7)(a+2) negative 14 and positive a

4a-7= 0 a +2 = 0

 

a=7/4 a=-2 Sub a=e^x

 

e^x = 7/4 and e^x= -2 Take ln of both sides

xlne=ln 7/4 and xlne=ln-2

x=ln 7/4 and x=ln-2

x cannot equal (-) log answer is x=ln7/4