Recall that we are given a function and we are told to apply some transformations to it. First, let us forget about the given function and just call it f.
If we want to apply a vertical translation, we simply do the following
[tex]f+D[/tex]
This transformation would move the function f a total of D units. If the sign of D is positive, it would move D units up, and it would move D units down otherwise.
Now, if we have a function f and apply a vertical stretch, we simply multiply the function by the stretching factor as follows
[tex]K\cdot f[/tex]
where K is the stretching factor.
Then, we start with our function f. We are told that we will apply a vertical translation of 2 units down. Then D=-2. So we get
[tex]\log _5x\text{ -2}[/tex]
Now, to this function we need to apply a vertical stretch of factor 8. That is, take K=8. So we get that our final function is
[tex]8\cdot(\log _5x\text{ -2) = 8}\cdot\log _5x\text{ -8}\cdot2\text{ = 8}\cdot\log _5x\text{ -16}[/tex]
Then the final function would be
[tex]8\cdot\log _5x\text{ -16}[/tex]
