Grade 11 Math Problem, cant figure out the transformations

[baller2353]

1. The problem statement, all variables and given/known data
the mother graph is y = 2 ^ x


2. Relevant equations
y=2 ^ x



3. The attempt at a solution
so i know the graph is flipped upsidedown and the whole graph is moved up 6 spots
so i can get y = -2 ^ x + 6

however the solutuion is y =-2 ^2x +6.
i cant seem to figure out how they got the exponent to the exponent 2x. I dont see what horizontal transformations could give you this.

question is in the picture attached.
thank u for your time.

[Mentallic]

1. The problem statement, all variables and given/known data
the mother graph is y = 2 ^ x


2. Relevant equations
y=2 ^ x



3. The attempt at a solution
so i know the graph is flipped upsidedown and the whole graph is moved up 6 spots
so i can get y = -2 ^ x + 6

however the solutuion is y =-2 ^2x +6.
i cant seem to figure out how they got the exponent to the exponent 2x. I dont see what horizontal transformations could give you this.

question is in the picture attached.
thank u for your time.

Notice that for y=-2^x+6, it would cut the x-axis at x=\log_26=\frac{\ln6}{\ln2}\approx 2.58 but from the graph it looks like it's cutting the x-axis at about 1.2 or so, which is about half. Well, how do we get an answer of x=\frac{1}{2}\left(\log_26\right) ? Solving backwards we have
2x=\log_26
2^{2x}=6
2^{2x}-6=0

thus we needed an exponent of 2x. Now in an exam or even when solving other questions, you won't go into this much detail, all you need to do is look at where it approximately cuts the x-axis and then use a suitable integer scalar for the exponent (I doubt they'll ever use anything but integers, because it's just not detailed enough).