Question

What are the roots of `2^x = 3-x` ?

Answer 1

`x=1`

Explanation:

By observation `x=1` is a solution, since:

`2^color(blue)(1) = 2 = 3 - color(blue)(1)`

Note also that:

• `2^x` is monotonically increasing with `x`

• `3-x` is monotonically decreasing with `x`

Hence they intersect at just one point - the one we found.

graph{(y-2^x)(y-3+x) = 0 [-9.92, 10.08, -2.52, 7.48]}

Footnote

Equations of the form: `a^x = bx+c` are not easy to solve in general. Unless there is an "obvious" solution then you are usually left to find numerical approximations or use the Lambert W function (which is actually a family of functions).

Answer 2

`x = 1`

Explanation:

Making `y = 3-x` we have

`2^(3-y)=y` or `2^3 = y 2^y` At this point `y = 2` satisfies the equation because

`2^3=2 xx 2^2 = 2^3`

and then

`2 = y=3-x rArr x = 1`

NOTE:

`y 2^y` is a strict increasing function for `y ge 0` so it crosses only once the value `2^3` which states the solution uniqueness.