Question

What fuction satisfies the following coordinates? `(0,1)(52,2)(104,4)(156,8)(208,16)(260,32)(312,64)` and so on?

Answer 1

`x÷52=(ln (y))÷(ln (2))`

Explanation:

`x` values are
Multiples of `52` starting from 0,1,2,3,4,5,6,..
`y` values are
Powers of 2 starting from
`0,1,2,3,4,5,6`,...
Thus `x=52a`
`a=x÷52`
where `a=(0,1,2,3,4,5,6,...)`
While `y=2^a` where `a=(0,1,2,3,4,5,6,...)`
Simplifying
`a=log_2 (y)`
By change of base rule that `log_a(b)=log_c(b)/(log_c(a))`
`log_2(y) = ln (y)÷ln (2)` We have set `c` as `e`.
Now,
`a=ln (y)÷ln (2)`
Equating a from the expressions
`x÷52=(ln (y))÷(ln (2))`

Answer 2

`y = 2^(x/52)`

Answer 3

The answer is `y=2^(x/52)`

Explanation:

Let 's make a table

`color(white)(aaaa)` `n` `color(white)(aaaa)` `0` `color(white)(aaaaa)` `1` `color(white)(aaaaaa)` `2` `color(white)(aaaaa)` `3` `color(white)(aaaaaa)` `4` `color(white)(aaaaaa)` `5` `color(white)(aaaaaa)` `6`

`color(white)(aaaa)` `x` `color(white)(aaaa)` `0` `color(white)(aaaa)` `52` `color(white)(aaaa)` `104` `color(white)(aaaa)` `156` `color(white)(aaaa)` `208` `color(white)(aaaa)` `260` `color(white)(aaaa)` `312`

`color(white)(aaaa)` `y` `color(white)(aaaa)` `1` `color(white)(aaaaa)` `2` `color(white)(aaaaaa)` `4` `color(white)(aaaaaa)` `8` `color(white)(aaaaa)` `16` `color(white)(aaaaa)` `32` `color(white)(aaaaa)` `64`

From the table, we can see that

`y=2^n`, `AA n in NN`

and

`x=26xx2n`

Eliminating `n` from the `2` equations,

`n=x/52`

`y=2^(x/52)`