Question

Why are the `x`-coordinates of the points where the graphs of the equations `y=4^-x` and `y=2^(x+3)` intersect solutions of the equation `4^-x=2^(x+3)`?

Answer 1

`x=-1`

Explanation:

Note that `4=2^2` and that we can use the rule `(x^a)^b=x^(ab)` and so we can write:

`4^(-x)=2^(x+3)`

as

`(2^2)^(-x)=2^(x+3)`

`2^(-2x)=2^(x+3)`

And so we can equate the two exponents:

`-2x=x+3`

`-3x=3`

`x=-1`

And now let's check the solution:

`4^(-x)=2^(x+3)`

`4^(-(-1))=2^((-1)+3)`

`4^1=2^2=4`

Answer 2

Please consider what we mean by intersection; we mean a condition where the two equations have the same `(x,y)` values. This implies that point in equation [1] `(x_1,y_1) = (x_2,y_2)` in equation [2]

Explanation:

Therefore, you can find the value where `x_1 = x_2` by setting one expression that is y in terms of x equal to the other expression that is y in terms of x and then solving for x.

Similarly, you can find the value where `y_1 = y_2` by setting one expression that is x in terms of y equal to the other expression that is x in terms of y and then solving for y.

It is a happy situation where we can start with the two equations.

`y=4^-x" [1]"`
`y=2^(x+3)" [2]"`

and we can obtain two expressions for x terms of y:

`ln(y)=ln(4^-x)`
`ln(y)=ln(2^(x+3))`

`ln(y)=-xln(4)`
`ln(y)=(x+3)ln(2)`

Starting with equations [1] and [2], x in terms of y are equations [3] and [4]

`x = -ln(y)/ln(4)" [3]"`
`x= ln(y)/ln(2)-3" [4]"`

NOTE: We are lucky that we can do the above; this is not always possible.

To find where the where the two equations intersect, we set x in terms y equal to x in terms of y and the solve for y:

#-ln(y)/ln(4) = ln(y)/ln(2)-3

`ln(y)(ln(2)+ln(4)) = 3`

`ln(y)ln(8) = 3`

`ln(y) = 3/ln(8)`

`y = e^(3/ln(8)) larr` this is the y value where the two equations intersect.

Is summary, we usually express y in terms of x because we have agreed on that and this makes the x coordinate the easiest to find. But, if we can express x in terms of y, then there is no reason why cannot choose to solve for y. However, there are some equations where one or the other form does not exist.

I hope that this answers your question.