log help

i don't understand why some log equations substitute y into the equations

for example,

log5(x) + 6 logx (5) = 5
log5(x) + 6/log5(x) = 5
let y = log5(x)
so y + 6/y = 5....

or

3^(2x+1) + 5 = 16(3^x)
let y = 3^x
so 3y^2 + 5 = 16y ...

why are we allowed to do that in some cases? how do i know when i should use that technique? i hope my question is clear enough, thanks in advance

It's just a technique you can use to help you solve these kinds of equations. You can use it any time you like if it's useful.

E.g. if you had $3^x + 5^x = 3$ then using $y=3^x$ won't be much help because you'll be left with

$y + 5^x = 3$


But if you have something like $3^{2x+1} + 5 = 16(3^x)$ then this can be written as

$3\times (3^x)^2 + 5 = 16(3^x)$

Then if you use $y=3^x$ you get

$3y^2 + 5 = 16y$

This is a quadratic which you should find easier to deal with than the equation above it. You just have to remember to substitute back in $3^x$ at the end.

Does this make sense?

it does make sense but what would be an alternative way to solving it without turning it into a quadratic?