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

Question

What in the world is this? please help!!

Answer 1

(1) Can also be written e^x=e^-2x+1, e^3x=1+e^2x.

Let y=e^x, then y³=1+y², y³-y²-1=0.

Let f(y)=y³-y²-1, then f'(y)=3y²-2y. Newton's iterative Method:

y_n+1=y_n-f(y_n)/f'(y_n), where y₀=1 (arbitrarily).

y₁=1-(-1)/(1)=2; y₂=2-3/8=13/8; y₃=1.485...; y₄=1.465...; etc.

y stabilises at y=1.46557123188 approx.

e^x=1.46557123188, x=ln(1.46557123188)=0.382245 approx.

(2) Can also be interpreted e^x=e^-2(x+1), which is much easier to solve, because the exponents are equal:

x=-2x-2, 3x=-2, x=-⅔.