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 \(x=-2,y=\frac{1}{4}\); \(x = - 1,y=\frac{1}{2}\); \(x=0,y = 1\); \(x = 1,y=2\); \(x=2,y = 4\); \(x=3,y=8\). In the table for \(y=\log_{2}(x)\), when \(x=\frac{1}{4},y=-2\); \(x=\frac{1}{2},y=-1\); \(x = 1,y=0\); \(x=2,y = 1\); \(x=4,y=2\); \(x=8,y=3\). 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 function. As \(x\to+\infty,y\to+\infty\) and as \(x\to-\infty,y\to0\). The graph of \(y=\log_{2}(x)\) is a logarithmic function. As \(x\to+\infty,y\to+\infty\) and as \(x\to0^{+},y\to-\infty\). The graphs of \(y = 2^{x}\) and \(y=\log_{2}(x)\) are symmetric about the line \(y = x\)

Step3: Use exponentials to graph logarithms

Since \(y=a^{x}\) and \(y=\log_{a}(x)\) are inverse functions (\(a>0,a
eq1\)). 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 table for \(y = 2^{x}\) and \(y=\log_{2}(x)\) are swapped.
• For the graphs: The graphs of \(y = 2^{x}\) and \(y=\log_{2}(x)\) are symmetric about the line \(y=x\)
• For using exponentials to graph logarithms: Since exponential and logarithmic functions (\(y=a^{x}\) and \(y=\log_{a}(x)\)) are inverse functions, if we know the key points \((x,y)\) of the exponential function \(y=a^{x}\), then the key points of the logarithmic function \(y=\log_{a}(x)\) are \((y,x)\)