Solve for x: e^2x - e^x - 6 = 0

MarkSA

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Sep 8, 2007
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243
Solve for x with exact values (no decimals):

e^(2x) - e^x - 6 = 0

I know how to generally solve this problem - by changing it to quadratic form:
(e^x - 3)(e^x + 2)=0
And seeing that:
e^x = 3
and
e^x = -2 (though this one is not possible)

So i'm left with e^x = 3. Ln both sides:
xln(e) = ln(3)
x = ln(3)

But I can't do it this way since solving ln(3) would require a calculator (and a huge number of decimals no less)

Is there another trick to solving problems such as this for exact values? Thanks
 

ilaggoodly

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Oct 4, 2007
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thats not log you know, its ln
why not just leave it as x = ln (3) , the ln of 3 is a unique irrational number
thats as exact as saying x = sqrt 3 which is considered an exact answer
 

MarkSA

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Sep 8, 2007
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Sorry, I just noticed that mistake and changed it right before your post.

The instructions for this problem are to 'Give exact values only.' I was a little confused about what that meant when it was said in class (it's also mentioned on the sheet)... but I thought I understood it to mean that we weren't supposed to use decimals. I guess in fraction form?

Is there a way to solve this problem and get an answer as a fraction?
 

tkhunny

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You're missing the point. Let's talk about \(\displaystyle \L\pi\).

3.14 -- NOT an exact value.
3.14159265358979323846264338 -- NOT an exact value.
22/7 -- NOT an exact value.

No matter how many decimal places, or how big the rational number in the fraction, these are approximations ONLY.

The ONLY particularly convenient way to express the EXACT value of \(\displaystyle \L\pi\) is ... are you ready ... prepare your self for a great revelation ... \(\displaystyle \L\pi\)! That's it. No, really. There are a few other ways to express this value exactly, but you do not want to know those, yet.

The EXACT value of ln(3) is ... guess what...ln(3). That's it. No rational number fraction is closer than that. No finite decimal expansion is closer than that. Just ln(3). It's a number. Numbers are our friends. Don't deny some of them their existence.
 

MarkSA

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Sep 8, 2007
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Okay, that makes me feel better. Hopefully my teacher won't deny the existence of it too :)

I've got one more similar type of question:

22^(1.5x) = 13

The trick I used in another problem like this was to get the coefficients (if that's the correct term), which are 22 and 13 in this case, to be the same, and then drop them and solve the equation with the exponents. Doing that in a previous problem actually gave me a nice little fraction for an answer.

It's a bit more difficult (impossible?) to do that in this one though I think. I suppose I could just log them:
1.5xlog22 = log13
1.5x = log13/log22
x = 2log13/3log22

That's another calculator operation though, and not a nice one like ln3 either. Is that basically as simplified as that answer can get too?
 

o_O

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Oct 20, 2007
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Just as with the answer to your first question, you won't get an exact value other than what you have expressed in logarithms. log13 and log22 can't be expressed in fractions. However, you can simplify your answer a bit:

\(\displaystyle log_{a}b = \frac{log_{c} b}{log_{c} a}\) where c could be any number just as long as you are consistent in the bases in the numerator and the denominator. In this case, in your second question, c = 10.

So your answer ends up looking "nicer": x = \(\displaystyle \frac{3}{2} log_{22}13\)
 

MarkSA

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Sep 8, 2007
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243
Oh! Good catch there. Thanks.
 
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