it was derived as follows:
you know that 2^2 = 4
you want some value of b^(-2) to be also equal to 4
you start with b^(-2) = 4
take the log of both sides of the equation to get:
log(b^(-2)) = log(4)
since log(b^a) = a*log(b), that equation becomes:
-2*log(b) = log(4)
divide both sides of that equation by -2 to get:
log(b) = log(4)/(-2)
evaluate that equation to get:
log(b) = .6020599913/(-2) = -.3010299957
by the basic law of logarithms that says y = log(x) if and only if 10^y = x, you get:
log(b) = -.3010299957 if and only if 10^-.301029997 = b
solve for b to get b = .5 which is equal to 1/2.
you wound up with b^(-2) = 4 if and only if b = 1/2.
your solution is therefore that y = (1/2)^x.
here's the graph of y = 2^x and y = (1/2)^x
you can see that these two equations are reflections of each other about the y-axis.
this may have been the hard way to derive it.
usually you just replace x with -x and you should get the equivalent equation that's a reflection of the original equation about the y-axis.
start with y = 2^x
replace x with (-x) to get:
y = 2^(-x)
that's the same as y = 1 / 2^x
since 1 to any power is equal to 1, that equivalent to y = 1^x / 2^x which is the same as y = (1/2)^x.
the solution was derived in two different ways.
i think the second way was easier.