Question

e^x - e^-x/e^x+e^-x = 1/2

Answer 1

\[(e^x-e ^{-x})/(e^x+e ^{-x})=0.5\]

Answer 2

Left hand side = tanh x (hyperbolic tangent)

Answer 3

@Henryblah how do I solve the rest?

Answer 4

You can rewrite tanh x as e^(2x)-1 / e^(2x) +1
Multiply out, simplify then take logs.

Answer 5

@estudier where did you get 2x from?

Answer 6

e^x + e^-x ie e^x + 1/e^x Add them

Answer 7

A little more assistance here please.

Answer 8

\(\LARGE e^x + \frac{1}{e^x} =\frac{e^{2x} + 1}{e^x}\quad \text{make sense?}\)

Answer 9

that's the denominator ^

Answer 10

\(\Large e^x - \frac{1}{e^x} = \frac{e^{2x} - 1}{e^x} \leftarrow \text{numerator}\)

Answer 11

@math102 are you understanding so far?

Answer 12

So we are now subtracting 1/e^x because I see now you have -1/e^x

Answer 13

yepp

Answer 14

so if we rewrite \(\Large \frac{e^x - e^{-x}}{e^x + e^{-x}}\)

you'll have

\[\Huge \frac{\frac{e^{2x} - 1}{e^x}}{\frac{e^{2x} + 1}{e^x}}\]

right?

Answer 15

Okay.

Answer 16

cancel the \(\large e^x\) on both doth denominators and you'll get to where @estudier left off :D

Answer 17

Keep going...

Answer 18

uhmm well now we have

\(\LARGE \frac{e^{2x} - 1}{e^{2x} + 1} = \frac{1}{2}\)

if we cross multiply...

\(\LARGE 2e^{2x} - 2 = e^{2x} + 1\)

do you know how to go from there?

Answer 19

Okay so now I have e^2x -1/ e^2x + 1

Answer 20

yup then that \(\uparrow\)

Answer 21

\[e ^{2x}-1/ e ^{2x}+1\]

Answer 22

Okay so now what do I do with the 2x's?

Answer 23

nothing yet..continuing from what i said earlier..you will put the e^2x on one side and the constants on the opposite

Answer 24

So 2e^2x=2-1?

Answer 25

2e^2x - e^2x = 2 + 1

Answer 26

So I have 2e^2x-e^2x=3

Answer 27

and what is 2e^2x - e^2x?

Answer 28

So I dont use the 3 on the right side of the equation?

Answer 29

no..there's a 3..im asking you to simplify 2e^2x - e^2x

Answer 30

Would it be 1e^2x?

Answer 31

yup! but 1 is no longer needed to write

so we have e^2x = 3

take the ln of both sides

Answer 32

How do I take the natural log of e^2x?

Answer 33

let's put it as ln e^2x for now

Answer 34

So e ln 2?

Answer 35

uhmm no...wait.. you still have equal 3

ln e^2x = ln 3

do a power rule on ln e^2x

Answer 36

I tbink I get it now 2x= ln 3
x=ln 3/2
x=.5493?

Answer 37

After e^2x=3 should I have written 2x ln e =3?

Answer 38

uhmm lemme write it

ln e^2x = ln 3

2x ln e = ln 3

2x (1) = ln 3

x = (ln 3)/2

Answer 39

What happened to the 2x(1) in the third step you wrote?

Answer 40

Nevermind I got it. Thank you so much. You've been more than helpful. Superb!

Answer 41

no biggie :D