Question

`8x=4^x`.. Find x?

Answer 1

There are two solutions: `x=2, x~=0.18`

Explanation:

Most of the time, questions of this nature are quite difficult. However, by observation, we can see that when `x=2`:

`8(2)=16=4^2`

Is it possible that there is more than one solution? Yes.

And so another way to approach these kinds of questions is to graph the right side and the left side separately:

graph{(y-8x)(y-4^x)=0[-2,5,-5,20]}

By the graph, we can see that there is a second solution that is approximately `x=0.18`

Answer 2

`x = {0.15495346619034528, 2}`

Explanation:

Introducing the Lambert function `W`

https://es.wikipedia.org/wiki/Funci%C3%B3n_W_de_Lambert

we have

`X = Y e^Y hArr Y = W(X)`

so

`8x = 4^x rArr 4* 2x=2^(2x)`

Making now `xi=2x` we have

`4 xi = 2^xi = (e^(log 2))^xi = e^(xi log 2)` or

`1/4 = xi e^(-xi log 2)`

multiplying both sides by `-log 2` we have

`-log 2/4 = -(xi log 2)e^(-xilog 2)`

now making `X=-log 2/4` and `Y = -xi log 2` we have

`-xi log2=W(-log 2/4)` or

`xi = -1/log 2 W(-log 2/4) = 2x` then

`x = -1/(2 log 2)W(-log 2/4) = (0.15495346619034528, 2)`