The given logarithmic function [tex]\(f(x)=\log_5(x+4)-2\)[/tex] undergoes a transformation of a horizontal shift to the left by 4 units and a vertical shift downward by 2 units.
The given function is [tex]\(f(x) = \log_5(x + 4) - 2\)[/tex]. The transformation applied to this function involves both horizontal and vertical shifts. The term (x + 4) inside the logarithm signifies a horizontal translation to the left by 4 units. This implies that each x-coordinate in the original function is replaced by x + 4, causing the graph to shift leftward.
Simultaneously, the constant term -2 outside the logarithm results in a vertical translation downward by 2 units. Subtracting 2 from the entire function vertically displaces each y-coordinate. Consequently, the entire graph shifts downward.
In summary, the transformation can be succinctly described as a composite effect: a horizontal translation to the left by 4 units, induced by (x + 4), and a vertical translation downward by 2 units, prompted by -2. These shifts modify the position of each point on the graph, providing a clear picture of how the function f(x) is altered from its original state [tex]\(f(x) = \log_5(x)\)[/tex]. The resulting function exhibits a leftward relocation and downward adjustment, as specified by the transformation parameters.