Problem 77 - Algebra A Combined Function

Question

Graph each logarithmic function. $$ y=\log _{3} x $$

Step-by-Step Solution

Answer

Plot key points and sketch the curve with a vertical asymptote at \(x = 0\).

Step 1: Understand the Basic Function

The function \(y = \log_3 x\) is a logarithmic function with base 3. It represents the inverse of the exponential function \(y = 3^x\). Understanding its properties helps in sketching the graph.

Step 2: Identify Key Points

To graph \(y = \log_3 x\), calculate a few key points:- When \(x=1\), \(y=\log_3 1=0\).- When \(x=3\), \(y=\log_3 3=1\).- When \(x=\frac{1}{3}\), \(y=\log_3 \frac{1}{3}=-1\). These points will help in plotting the curve.

Step 3: Plot the Key Points

On a coordinate plane, plot the points - \((1, 0)\)- \((3, 1)\)- \((\frac{1}{3}, -1)\).These points show where the graph will pass through as it curves.

Step 4: Sketch the Graph

Draw a smooth curve through the points plotted in Step 3. The graph approaches the y-axis (asymptote) but never touches it, and it rises as \(x\) increases.

Step 5: Note the Characteristics

The graph of \(y = \log_3 x\) has the following features:- It passes through the point \((1, 0)\).- The x-axis is an asymptote, meaning the graph approaches it but never touches or crosses it.- The graph increases slowly as \(x\) gets larger.

Key Concepts

Graphing Logarithms

Graphing logarithmic functions can initially seem challenging, but by following a structured approach, it becomes easier. At the heart of any logarithmic function is the concept that it is an inverse of an exponential function. For example, the function \( y = \log_{3}x \) corresponds to the exponential function \( y = 3^x \). This inverse relationship plays a key role in understanding the graph's shape and properties.

When beginning to graph this logarithmic function, you'll want to identify key points that will serve as the framework for your graph. These usually include points where \( x \) equals 1, the base of the logarithm, and fractions related to the base. For instance:

• When \( x=1 \), \( y=\log_{3}1=0 \), translating to the point \((1, 0)\).
• When \( x=3 \), \( y=\log_{3}3=1 \), known as the point \((3, 1)\).
• When \( x=\frac{1}{3} \), \( y=\log_{3} \frac{1}{3}=-1 \), pinpointing \((\frac{1}{3}, -1)\).

These key points will guide the overall shape of your graph. Connecting these points with a smooth curve will reveal the logarithm's characteristic near-y-axis approach, showing its asymptotic behavior, and its slow, steady rise as \( x \) grows larger.

Inverse Functions

Inverse functions are fundamental in mathematics because they reverse the roles of inputs and outputs. For logarithmic functions, their relationship with exponential functions defines them as inverses. For the given function \( y = \log_3 x \), its inverse can be expressed as \( y = 3^x \). This means that for each value of \( y \) in the logarithmic function, there is a corresponding \( x \) in the exponential function.

Visualizing inverse functions is key to understanding their properties. When graphed, a logarithmic function and its inverse (the exponential function) reflect each other over the line \( y = x \). This line of symmetry underscores their respective properties as inverses.

• Exponential functions extend from the lower left to the upper right, with a rapid rate of increase as \( x \) rises.
• Logarithmic functions move from lower left to upper right as well but start slow and only gradually incline.

This mirror-like behavior is at the core of methods like solving equations where equations of exponential growth and logarithmic scaling are simplified through inversion methods.

Properties of Logarithms

The properties of logarithms are crucial tools for solving equations and simplifying expressions. Understanding these properties makes handling complex logarithmic problems much more straightforward. Here are some of the core properties to keep in mind:

• Product Property: \( \log_b(MN) = \log_bM + \log_bN \). It expresses that the log of a product is the sum of the logs.
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_bM - \log_bN \). It expresses that the log of a quotient is the difference of the logs.
• Power Property: \( \log_b(M^k) = k\log_bM \). This property helps in dealing with logarithms that have exponents by letting you bring the exponent out front as a multiple.

These properties provide powerful techniques to reconfigure and simplify logarithmic equations, often transforming them into forms that are much easier to work with or further analyze. Delight in the elegance and utility they bring to mathematical problem-solving!