How is the graph y=log (2x) + 3 related to the graph of y=log (x)

The basic shapes of the 2 graphs will be very similar.

Start with y = log (x). This graph begins in Quadrant IV and ends in Q I. It has no y-intercept and no x-intercept, but does have horiz. intercept (1,0).

The graph of y=log (2x) looks the same as that of y = log (x), except that the first graph is compressed horizontally. Also, this horizontally-stretched graph has to be translated upward 3 units to obtain the graph of y=log (2x) + 3.