Problem 45 - Intermediate Algebra

Question

Graph \(f(x)=\log _{3} x\) by reflecting the graph of \(g(x)=3^{x}\) across the line \(y=x\).

Step-by-Step Solution

Answer

Graph \(f(x)=\log_{3}x\) by reflecting \(g(x)=3^x\) across \(y=x\).

Step 1: Identify the Basic Function

The function given is \(f(x) = \log_{3} x\). This function is the logarithmic function with base 3. Our goal is to graph it using a reflection method.

Step 2: Understand Reference Function

To graph \(f(x) = \log_{3} x\), it's helpful to understand its inverse function \(g(x) = 3^x\), which is an exponential function. They are related such that \(f(x)\) is the reflection of \(g(x)\) across the line \(y=x\).

Step 3: Plot Key Points of Exponential Function

First, plot the points of \(g(x) = 3^x\). Choose a few values of \(x\) to calculate \(y\). For example, when \(x=0\), \(y=1\), when \(x=1\), \(y=3\), and when \(x=2\), \(y=9\).

Step 4: Reflect Points Across the Line \(y=x\)

To reflect these points across \(y=x\), swap the \(x\) and \(y\) coordinates. For example, the point \((0,1)\) becomes \((1,0)\), \((1,3)\) becomes \((3,1)\), and \((2,9)\) becomes \((9,2)\).

Step 5: Graph the Logarithmic Function

Using the reflected points, graph \(f(x) = \log_{3} x\). Include the points \((1,0)\), \((3,1)\), and \((9,2)\). Ensure the function is a smooth curve approaching the \(y\)-axis (which is the vertical asymptote) from the right.

Key Concepts

Graphing Techniques

When graphing logarithmic functions, like \(f(x) = \log_{3} x\), a systematic approach often involves understanding their relationship with exponential functions. You begin by reflecting the corresponding exponential function graph around the line \(y=x\). This exploration of reflections helps visualize how closely linked logarithmic and exponential functions are.

Here’s a simplified process:

• Start by understanding the basic shape and behavior of the exponential function, which is \(g(x) = 3^x\).
• Plot key points of \(g(x) = 3^x\) on the graph. For instance, points like \((0,1)\), \((1,3)\), and \((2,9)\) show how the function grows.
• To graph the logarithmic function \(f(x) = \log_{3} x\), reflect these points across the line \(y = x\). This means swapping each point's \(x\) and \(y\) values.

The resulting graph of \(f(x) = \log_{3} x\) will have a shape that approaches the vertical asymptote but never quite touches it, resembling the behavior seen in exponential growth but mirrored.

Inverse Functions

Understanding inverse functions is key to grasping the relationship between logarithmic and exponential functions. An inverse function essentially "reverses" the operation conducted by the original function. In this case, the inverse of \(f(x) = \log_{3} x\) is \(g(x) = 3^x\). This relationship is visualized by reflecting the exponential function graph across the line \(y = x\).

Steps to understand inverse relationships:

• The line \(y = x\) is crucial as it acts as the mirror which illustrates the reflection and thus, the inversion.
• By swapping the coordinates \((x, y)\) of the exponential function, the points become those of the logarithmic function.
• The point \((1, 0)\) acts like an anchor on this graph, showing that 3 raised to 0 is indeed 1, corresponding with the zero log at base 3.

Think of inverse functions like undoing a math operation: logarithms undo the multiplication done by exponentials. This is why understanding their inverse nature is so intuitive when learning about these functions.

Exponential Functions

Exponential functions like \(g(x) = 3^x\) are foundational in understanding logarithms and their graphs. Recognizing how exponential functions create a curve that rises rapidly helps us see why logarithms have their characteristic shape. As you progress along the \(x\)-axis, an exponential function's output climbs steeply, illustrating its defining feature: exponential growth.

Some principles of exponential functions:

• They are of the form \(g(x) = a^x\) where \(a\) is a constant; here, \(a = 3\).
• The graph typically passes through the point \((0, 1)\) because any number raised to the power of zero equals one.
• For each unit increase in \(x\), the value of \(g(x)\) multiplies by the base \(a\) (e.g., moving from \((0, 1)\) to \((1, 3)\)).

Recognizing these features aids in understanding their logarithmic counterparts. Logarithmic graphs move in the opposite manner, slowing down their rise along the axes as seen upon reflection. This interplay paints a comprehensive picture of both mathematical concepts.