Answer:
First, let's explain the transformations in a general way:
Vertical shift.
If we have a function f(x), a vertical shift of N units is written as:
g(x) = f(x) + N
This will move the graph of f(x) up or down a distance of N units.
if N is positive, then the shift is upwards
if N is negative, then the shift is downwards.
Horizontal shift.
If we have a function f(x), a horizontal shift of N units is written as:
g(x) = f(x + N)
This will move the graph of f(x) to the right or left a distance of N units.
if N is positive, then the shift is to the left
if N is negative, then the shift is to the right.
Horizontal dilation/contraction.
For a function f(x), an horizontal contraction dilation is written as:
g(x) = f(k*x)
where:
k is called the "scale factor"
If k < 1, then the graph "dilates" horizontally.
if k > 1, then the graph "contracts" horizontally.
Now, in this problem we have:
f(x) = log(x)
And the transformed function is:
g(x) = f(0.25*(x - 5)) + 3
Then, the transformations that take place here are, in order:
Vertical shift of 3 units up:
g(x) = f(x) + 3.
Horizontal shift of 5 units to the right:
g(x) = f(x - 5) + 3
Horizontal dilation of scale factor 0.25
g(x) = f( 0.25*(x - 5)) + 3
replacing f(x) by log(x) we have
g(x) = log(0.25*(x - 5)) + 3.
