Question

what do you notice about the tables for 2^x and log_2x?
they are flipped all the tables are backwards
what do you notice about the points graphed for each function?
each points are going to + infinte.
how can we use our understanding of exponentials to help us graph logarithmic
functions?

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 \((y,x)\)