E to the power of x SUBTRACT E to the power of -x divided by 2 equals 8 what is x?

Answer 1

(ex-e-x)/2=8

ex-e-x=16

ex = 16-e-x

ln(ex) = ln(16-e-x)

x = ln(16-e-x)

x = ln((16ex-1)/ex)

x = ln(16ex-1) - ln(ex)

x = ln(16ex-1) - x

2x = ln(16ex-1)

e2x = eln(16e^x - 1)

e2x = 16ex-1

e2x-16ex +1 = 0

Consider ex as a whole to be a dummy variable "u". (ex=u) The above can be rewritten as:

u2-16u+1=0

u2-16u=-1

Using completing the square, we can solve this by adding 64 to both sides of the equation (the square of one half of the single-variable coefficient -16):

u2-16u+64=63

From this, we get:

(u-8)2=63

u-8=(+/-)sqrt(63)

u=sqrt(63)+8, u= 8-sqrt(63)

Since earlier we used the substitution u=ex, we must now use the u-values to solve for x.

sqrt(63)+8 = ex

ln(sqrt(63)+8)= ln(ex)

x = ln(sqrt(63)+8) ~ 2.769

8-sqrt(63) = ex

ln(8-sqrt(63)) = ln(ex)

x = ln(8-sqrt(63)) ~ -2.769

So, in the end, x~2.769 and x~-2.769. When backsubbed back into the original problem, this doesn't exactly solve the equation. Using a graphing calculator, the solution to this equation can be found to be approximately x=2.776 by graphing y=ex-e-x and y=16 and using the calculator to find the intersection of the two curves. This is pretty dang close to our calculated value, and rounding issues might account for this difference. The calculator, however, suggests that x~-2.769 is not a valid solution. This makes sense, and in fact it isn't a valid solution if you look at the graphs. This is an extraneous answer.