Question

Solve the equation
e^x+9e^-x/2=3
What is the solution in terms of natural logarithms?
x=
What is the decimal approximation for the solution?
x=

Answer 1

It depends on what you mean. If you mean \(\dfrac{e^{x} + 9e^{-x}}{2} = 3\), then you should write that. You definitely have not.

Answer 2

Yes, that is what I meant sorry...

Answer 3

Do you now why that is not what you have written?

Answer 4

Refer to the attachment generated by Wolfram Alpha

Answer 5

Since I was a little crabby, I'll give you one for free...

\(\dfrac{e^{x} + 9e^{-x}}{2} = 3\)

2 is never zero. Multiply by it.

\(e^{x} + 9e^{-x} = 6\)

\(e^{x}\) is never zero. Multiply by it

\(e^{2x} + 9 = 6e^{x}\)

Rearrange a little.

\(e^{2x} - 6e^{x} + 9 = 0\)

Recognize this as a quadratic-looking structure and simply factor it.

\((e^{x} - 3)^{2} = 0\)

Can you do the rest?

Answer 6

So now simplify it?

Answer 7

Is that the solution? How do I make it into a decimal?

Answer 8

You cannot simply that. We're solving aren't we? Do we have a value for 'x'? That would be a solution.

Answer 9

If you were solving a quadratic equation and encountered this, \((x-3)^{2} = 0\), what would you do?

Answer 10

I am guessing some how put it into decimal form

Answer 11

You're saying that if you factor a quadratic equation, and get \((x-3)^{2} = 0\), you cannot solve it? x = What?

Answer 12

Since I am having to put it into decimal terms and find the solution

Answer 13

3

Answer 14

How did you get that?

Answer 15

Solved the equation for x

Answer 16

Is it wrong

Answer 17

Given \((x-3)^{2} = 0\), we conclude that \(x - 3 = 0\) or \(x = 3\).

Is that what you mean by "solved the equation"?

Answer 18

yes

Answer 19

How would I put it into decimal form though?

Answer 20

Perfect. Now, do EXACTLY the same thing with \((e^{x} - 3)^{2} = 0\)

Answer 21

decimal approximation**

Answer 22

We're not there, yet. Let's find the EXACT solution.

Answer 23

Remember that quadratic equation. Apply that EXACT technique to this equation.

Answer 24

x=ln(3)

Answer 25

or 1.9086

Answer 26

It helps a conversation if you go one step at a time.

Very good. The REAL trick here was to recognize the Quadratic Form and treat it like that.

Answer 27

Ok so which one is the exact solution and do I put 1.9086 for the decimal

Answer 28

What is not exact about ln(3)?