Question

How do you solve `5^(x+3) <= 2^(x+4)`?

Answer 1

Replace `5` with `e^(ln(5))` and `2` with `e^(ln(2))`; then solve using the new exponents (ignoring the identical base values):
`color(white)("XXXX")` `x<= -2.24353`

Explanation:

Let `k = ln(5) =1.609438` and
let `m = ln(2) = 0.693147`

`5 = e^k` `color(white)("XXXX")`and`color(white)("XXXX")` `2 = e^m`

`5^(x+3) <= 2^(x+4)`
`color(white)("XXXX")`is equivalent to
`(e^k)^(x+3) <= (e^m)^(x+4)`

`kx+3k <= mx+4m`

`(k-m)x <= (4m-3k)`
`color(white)("XXXX")`since `k>m` we can divide both sides by `k-m`
`color(white)("XXXX")`without changing the orientation of the inequality

`x <= (4m-3k)/(k-m)`

Substituting the value of `ln(5)` for `k`
and the value of `ln(2)` for `m`
then pushing it all through my calculator...

I get `x <= -2.24353`