SOLUTION: Solve the following algebraically: e^x-6e^-x=1; this is what I have thus far. e^x(e^x-6e^-x)=e^x(1) e^2x-6=e^x e^2x-e^x-6=0 then use a quadratic equation to solve: -(-1

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve the following algebraically: e^x-6e^-x=1; this is what I have thus far. e^x(e^x-6e^-x)=e^x(1) e^2x-6=e^x e^2x-e^x-6=0 then use a quadratic equation to solve: -(-1      Log On


   

Question 115137: Solve the following algebraically: e^x-6e^-x=1; this is what I have thus far.
e^x(e^x-6e^-x)=e^x(1)
e^2x-6=e^x
e^2x-e^x-6=0
then use a quadratic equation to solve:
-(-1) +-square root (-1)^2-4(1)(-6)/2(1)
1+-square root 1+24/2
1+-square root 25/2
1+-5/2 = 1.0986
Is this correct?
Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You're on the right track. But remember u=e%5Ex which means x=ln%28u%29. Since you cannot take the log of a negative number, the only value that will work is u=%281%2B5%29%2F2=6%2F2=3

So our only answer is x=ln%283%29 which is approximately x=1.0986122886681

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Correct, but you made a leap of faith at the very end. You had:
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1+-5/2 = 1.0986
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and where did you get that from? The two answers you got from the quadratic equation are:
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(1+5)/2 = 3 and (1-5)/2 = -4/2 = -2
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How did you get the 1.0986???? The answer is that you solved for e^x = 3 and e^x = -2. You
can handle those in the same manner as below. I solved the problem by factoring which isn't
a whole bunch better than your way, but it is a little different. Maybe you'll find it a
little easier.
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Begin factoring at the point where you got:
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e^2x-e^x-6=0
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Think of this as:
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(e^x)^2 - (e^x) - 6 = 0
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Factor to get:
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(e^x -3)(e^x + 2) = 0
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This will be true if either of the two factors is zero because multiplication by zero on
the left side makes the left side equal to the right side.
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So either (e^x - 3) = 0 or (e^x + 2) = 0 will satisfy the equation. Let's do e^x - 3 = 0 first:
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Add + 3 to both sides to get:
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e^x = + 3
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Take the natural log (ln) of both sides:
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ln(e^x) = ln(3)
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Bring the exponent out as a multiplier:
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x*ln(e) = ln(3)
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Recognize that ln(e) = 1 which makes the equation become:
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x = ln(3) = 1.098612289
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This is the same as the answer you got.
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Next do the other factor:
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e^x + 2 = 0
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Subtract 2 from both sides:
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e^x = -2
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Take the ln of both sides:
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ln(e^x) = ln(-2)
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But you cannot take the ln of a negative number. So ignore this factor. The only answer
is x = ln(3).
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Hope this adds a little info to your knowledge of dealing with the base of the natural logarithms.
Good work on solving the problem with the quadratic equation. Just fill in the missing
step to show how you got 1.0986 from the answers you got from the quadratic equation.
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