solve equation

Question

solve e^2x- 4e^X+3=0

Answer 1

The easiest way to solve equations like this is to use the substitution  z =  e^x  .  With this, the equation becomes

 

  z² - 4 e³ z = 0  .  The solutions for z are

 

  z = 0  and z = 4e³   .   The z =0 solution is erroneous because there is no value of x which makes e^x = 0

 

 The other solution for x is ln(4 e³)  =  3 + ln(4).

 

 

If the problem had been  e^2x - 4 e^x + 3 = 0  , method of solution would have been similar.  After the substitution

 

  z²  - 4 z  + 3 = 0.    The solutions are  z = 1 and z = 3    giving the two values of x   as

 

   x = ln(1) = 0   and x =  ln(3)

 

 

Answer 2

Assuming Mark is right -

 

e ^(2x) - 4e^(x+3) = 0  -> add 4e^(x+3) to each side

e^(2x) = 4e^(x+3)        -> divide both sides by e^(x+3)

e^(2x-(x+3)) = 4           -> simplify the exponent

e^(x-3) = 4                   -> take the log of each side

x-3 = log(4)                   -> add 3 to each side

 

and the answer is

 

x = log(4)+3      

Answer 3

Jaun, the original equation is what it is in which is:

 

e^2x-4e^x+3=0

 

The way you could approach this problem is to treat the variable e^x as the unknown.

 

Let e^x =y

 

This would make the equation become:

 

y^2-4y+3=0

 

Now you got a simple quadratic equation to solve in which you could use the foil method.

 

(y-3)(y-1)=0

 

y=3 and y=1

 

This implies:

 

e^x=3 and e^x=1

 

Take the natural log of both sides of each equation to solve for x.

 

ln(e^x)=ln(3)  and  ln(e^x)=ln(1)

 

x=ln(3)  and x=ln(1)=0

 

Hope this helps.