Juan T.

asked • 11/12/14# solve equation

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

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## 3 Answers By Expert Tutors

The easiest way to solve equations like this is to use the substitution z = e

^{x}. With this, the equation becomes z

^{2}- 4 e^{3}z = 0 . The solutions for z are z = 0 and z = 4e

^{3}. 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 e

^{3}) = 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 z

^{2}- 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)

Dal J.

Thorough and accurate.

The only suggestion I have for Richard is to put a visual divider between the two different versions of the problem.

You had me convinced that I had made a mistake somewhere, rather than that I had only worked one of the possible problem statements...

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11/12/14

Dal J. answered • 11/12/14

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Expert Instructor in Complex Subjects and Public Speaking

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

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.

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Mark W.

11/12/14