How is the graph of y=4 (2) ^x+3 translated from the graph of y=4 (2) ^x?

Let's log 10 on both side of the equation for both of your equations

You'll get log y = (x-3) log 8

and log y = x log 8

Lets assume that a graph of log y against x is plotted. Then log y becomes the y variable, log 8 becomes the gradient and x becomes the x variable. To make visualizing easier, lets assume the gradient is m=log 8 = 1.

So the two log equation becomes,

y = x

and

y = x-3

Thus, from the x intercept we can see that the graph of y = 4 (2) ^x-3 is translated from the graph y=4 (2) ^x 3 units down.