What is the value of $a$ & $r$ when 2 points on the graph are given?

Question

I am working on algebraic functions and I am stuck on this problem:

$f(x) = a * r^x$
$(2,1),(3,1.5)$

This would be a simple problem if it weren't for that $1.5$ -

So, I have plugged in $2$ and $1$ into the function and this is what I got:

[I also solved for $3,1.5$]
So, this is where I get stuck every time.

I have tried solving it multiple ways but it always ends up to the same effect, where $r$ is always $=$ to something funny that is unsolvable - at least for me.

Could someone explain how you got your answer so that I can understand the process? I know how to do this type of problem, but it is just this one that I can't get!!!!

Thanks for your help and time!

EDIT: I need this simplified, that is why I cant leave $r^2 = 1.5$

Answer 1

So if we have the function f(x)=ar^x, then by substituting the first point given, namely $(2,1)$, we'll get:

$$1=ar^2$$

Similarly, substituting $(3, \frac{3}{2})$, we end up with

$$\frac{3}{2}=ar^3$$

Now, to find $a\neq 0$ and $r\neq 0 $, you should divide this two equations, what you get then?

Answer 2

$$f(2)=ar^2=1$$ $$a=\frac{1}{r^2}$$ $$f(3)=ar^{3}=1.5$$ $$r=1.5$$ $$a=\frac{1}{(1.5)^2}$$