natural logarithm?

Question

e^x + e^(-x) = 3

can you please work through that for me?

Answer 1

e^x + e^(-x) = 3

e^x + 1/e^x = 3

e^x * e^x / (e^x) + 1/e^x = 3

(e^x)^2 / e^x + 1/e^x = 3

(e^(2x) + 1 ) / e^x = 3

e^(2x) + 1 = 3 e^x

let y be e^x
y^2 + 1 = 3y

y^2 - 3y + 1 = 0

use quadratic formula
3 +/- sqrt(3^2 - 4(1)(1))
-------------------------------
............ 2


3 +/- sqrt(9 - 4)
--------------------
.............. 2


3 +/- sqrt(5)
-----------------
............ 2



y = e^x
( 3 + sqrt(5) )/2 = e^x
x = ln (3 + sqrt(5)/2)
x = .9624236501

3 - sqrt(5) / 2 = e^x
x = ln (3 -sqrt(5)/2)
x = -.9624236501

Answer 2

Let's introduce a new variable y = e^x. Then we can rewrite the original equation as follows:
y + 1/y = 3, let's multiply both sides by y
y^2 + 1 = 3y, now subtract 3y from both sides
y^2 - 3*y + 1 = 0.

Solving the above quadratic formula gives us two solutions:
y1 = (3+ sqrt(5)) / 2 and
y2 = (3- sqrt(5)) / 2.

Remember that y = e ^ x, therefore
e ^ x1 = (3+ sqrt(5)) / 2 and
e ^ x2 = (3- sqrt(5)) / 2.

Now resolve for x by applying natural logarithm to both sides:
x1 = ln((3+ sqrt(5)) / 2) = ln(3+ sqrt(5)) - ln (2) and
x2 = ln((3- sqrt(5)) / 2) = ln(3- sqrt(5)) - ln (2)

Answer 3

Let y=e^x

Then y+1/y= 3

y^2+1=3y

y^2-3y+1=0


y= 1/2{3+- Sqrt(9-4)}

=1/2 {3-Sqrt(5)} or 1/2{ 3+Sqrt(5)}
= 0.38196 or 2.6180
But y=e^x
therefore
x= natural log(0.38196) or ln(2.618)

= -0.9624 or 0.9624

Answer 4

e^x + e^(-x) = 3, let e^x= y
y + 1/y= 3 ( multiply by y )
y^2 -3y + 1 = 0
find the values of y ,...........
y= 2.618 , y = 0.382
e^x= y= 2.618 OR ,e^x= y = 0.382
x = ln 2.618=0.962
OR x = ln 0.382 = - 0.9623

Answer 5

e^x+1/e^x = 3, e^(2x) +1 = 3e^x , (e^(x))^2-3e^x+1=0

e^x= 1/2(3+/- sqrt(5)). x = ln{1/2(3+/-sqrt5)}

Answer 6

e^(2x) + 1 = 3.e^(x)
e^(2x) - 3.e^(x) + 1 = 0
Let y = e^x
y² - 3y + 1 = 0
y = [3 ± √5] / 2
y = 2.62 , y = 0.764
e^x = 2.62 , e^x = 0.764
x = ln 2.62 = 0.788
x = ln 0.764 = - 0.269

Answer 7

e^x + e^(-x) = 3
ln(e^x) + ln(e^-x) = ln 3
x ln e - x ln e =ln 3
x(ln e - ln e) = ln 3
x= ln 3
x= 1.09861