Exponential problems?

1) Multipluing the given problem, throughout by 2e^x,

==> 2(e^2x) + 2 = 5(e^x)
Let e^x = y;

==> 2(y^2) - 5y + 2 = 0; ==> (y-2)(2y-1) = 0
==> y = 2 or 1/2

==> e^x = 2 or 1/2

Taking log and solving, x = ln(2) or -ln(2)

Thus x = {-ln(2), ln(2)}. [ln stands for logarithm to base 'e']

I prefer to write as

e^x + 1/e^x - 2.5 = 0

Multiply both sides by e^x.

e^2x - 2.5e^x + 1 = 0

Note that e^2x = (e^x)^2.

Let u = e^x

u^2 - 2.5u + 1 = 0

Solve with the quadratic formula.

(2.5 +/- sqrt((2.5^2 - 4(1)(1))/2

The solutions are:
u = 2
u= 1/2

u = e^x = 2

So x = ln(2)

and

u = e^x = 1/2

So x = ln(1/2) = -ln(2)

Answer

x = +/-ln(2)

Let y = e^x

y + 1/y = 2.5

y ² + 1 = 2.5 y

y ² - 2.5 y + 1 = 0

y = [ - b ± √ ( b ² - 4 a c ) ] / 2 a

y = [ 2.5 ± √ ( 6.25 - 4 ) ] / 2

y = [ 2.5 ± √ ( 2.25 ) ] / 2

y = [ 2.5 ± 1.5 ] / 2

y = 2 , y = 1/2

e^x = 2 , e^x = 1/2

x = ln 2 , x = ln (1/2)

x = 0.693 , x = - 0.693

cosh(x) ≡ ½ [e^(x) + e^(-x)]
arcosh(x) ≡ ln [ x + √(x² - 1) ]

e^x + e^ (-x) = 2.5
½ [e^x + e^ (-x)] = 1.25
cosh(x) = 1.25
x = arcosh(1.25)
.. = ln [ 1.25 + √((1.25)² - 1) ]
.. = ± 0.69314718056

e^x+e^(-x)=2.5
e^x+1/e^x=2.5
e^x=a
(a^2+1)/a=2.5
a^2+1=2.5a
a^2-2.5a+1=0
(a-2)(a-.5)=0
a=2 or a=.5
e^x=2 or e^x=.5
x=ln(2) or x=ln(.5)
ln(2)=-ln(.5)
x=+/-ln(2)=+/-0.693147180559945309...

Since exp(x) is never 0, you can multiply through by exp(x) to get the equation:
exp(2x)+1=2.5*exp(x), or [exp(x)]^2 - 2.5[exp(x)] + 1 = 0.

You can then use the quadratic equation to solve for the values of exp(x) as:
(2.5 +/- sqrt(6.25-4))/2 = (2.5 +/- 1.5)/2 = {4/2,1/2}.

The two solutions are then exp(x)=2 and exp(x)=1/2,
or x = ln(2) and x = ln(1/2) = -ln(2)

Let's call x = ln(z)

so, e^(ln(z)) + e^(-ln(z)) = 2.5
z + 1/z = 2.5
z^2 + 1 = 2.5z
z^2 - 2.5z + 1 = 0

Hm. I'll solve this with the quadratic equation: (-b +/- Sqrt[b^2-4ac) / 2a
a=1, b=-2.5, c=1

So, z = (2.5 +/- Sqrt[6.25-4])/2 = (2.5 +/- Sqrt[2.25])/2 = (2.5 +/- 1.5)/2

So, z = 2 or z = 1/2.

x = ln(z), so x = ln(2) or x = ln(1/2)

e^x +1/e^x= 25/10 = 5/2
2e^2x +2 =5 e^x
2e^2x -5 e^x +2=0

let e^x =u
2u*2- 5u+2=0
u=( 5+- sqrt(25 -16))/4
u= (5+- 3)/4
u1= 2
u2= 1/2
e^x=2 , x1= ln 2
e^x= 1/2 ,x2= - ln2