what do you notice about the tables for 2^x and log_2x?\n they are flipped all the tables are backwards\n…

Explanation:

Step1: Analyze the tables

In the table for (y = 2^{x}), when (x=-2,y=\frac{1}{4}); when (x = - 1,y=\frac{1}{2}); when (x=0,y = 1); when (x=1,y=2); when (x = 2,y=4); when (x=3,y = 8). In the table for (y=\log_{2}(x)), when (x=\frac{1}{4},y=-2); when (x=\frac{1}{2},y=-1); when (x = 1,y=0); when (x=2,y=1); when (x=4,y=2); when (x=8,y=3). The (x) and (y) values of the two - function tables are swapped.

Step2: Analyze the graphs

The graph of (y = 2^{x}) is an exponential growth curve. As (x\to+\infty,y = 2^{x}\to+\infty) and as (x\to-\infty,y = 2^{x}\to0). The graph of (y=\log_{2}(x)) is a logarithmic curve. As (x\to+\infty,y=\log_{2}(x)\to+\infty) and as (x\to0^{+},y=\log_{2}(x)\to-\infty). The graph of (y = f(x)=2^{x}) and (y = f^{-1}(x)=\log_{2}(x)) are symmetric about the line (y = x)

Step3: Use exponentials to graph logarithms

Since (y=\log_{a}(x)) and (y=a^{x}) are inverse functions ((a = 2) in this case). If we know some key points ((x,y)) of the exponential function (y=a^{x}) (such as ((0,1),(1,a),( - 1,\frac{1}{a}))), then the key points of the logarithmic function (y=\log_{a}(x)) are ((y,x))

Answer:

• For the tables: The (x) and (y) values of the two - function tables are swapped.
• For the graphs: The graph of (y = 2^{x}) and (y=\log_{2}(x)) are symmetric about the line (y=x)
• For using exponentials to graph logarithms: Since (y=\log_{a}(x)) and (y=a^{x}) are inverse functions. If we know key points ((x,y)) of (y=a^{x}), then the key points of (y = \log_{a}(x)) are ((y,x))