Problem 26 - College Algebra Essentials

Question

Graph each inequality. $$y \geq \log _{3}(x-1)$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \geq \log_3(x-1)\) is the entire region that lies above the curve of \(y = \log_3(x-1)\), including the curve itself. The curve starts from the point (1,0) and extends to the right, upwards.

Step 1: Understand the Base Graph

The base graph is \(y = \log_3(x-1)\). This is a logarithmic function with base 3. It simply represents the exponent to which 3 must be raised to give x-1. Its domain will be \(x > 1\) since logarithm is undefined for zero and negative numbers.

Step 2: Draw the Base Graph

The graph of \(y = \log_3(x-1)\) will start from the point (1,0). For every integer x-value greater than 1, the y-value will be \(y = \log_3(x-1)\). It would be helpful to plot few points like (2, \(\log_3(2-1)\)) and (4, \(\log_3(4-1)\)) and then join them to get the curve.

Step 3: Apply the Inequality

Because the inequality is 'greater than or equal to' (\(\geq\)), the graph includes the original curve of \(y = \log_3(x-1)\). The inequality \(y \geq \log_3(x-1)\) is the region above this curve, including the curve itself. Hence, this upper region should be shaded on the graph to represent this inequality.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions, represented as \(y = \text{log}_b(x)\), play a significant role in many areas of mathematics and science. They are the inverses of exponential functions. When we write \(y = \text{log}_b(x)\), we're asking the question, 'To what exponent must we raise the base \(b\) to produce \(x\)?' For instance, \(\text{log}_3(9) = 2\) because \(3^2 = 9\). Understanding the characteristics of logarithms is key to graphing these functions and solving inequalities that involve them. The logarithmic equation \(y = \text{log}_b(x - h) + k\) features a horizontal shift of \(h\) and a vertical shift of \(k\). In the given exercise, \(y \text{ is greater than or equal to log}_3(x-1)\), represents values of \(y\) that are the exponent required to raise 3 to get \(x-1\) or greater. Remember, logarithms require the input \(x\) to be positive, as they are undefined for zero and negative values.

Graphing Inequalities with Logarithmic Functions

Graphing inequalities that involve logarithmic functions like \(y \text{ is greater than or equal to } \text{log}_3(x-1)\) can initially seem daunting, but it becomes manageable with practice. The graphing process starts with sketching the 'base' logarithmic function, without the inequality. For our example, we plot points on the graph of \(y = \text{log}_3(x-1)\). After plotting the curve, the inequality tells us which side of the base graph to shade. With an inequality such as \(y \text{ is greater than or equal to } \text{log}_3(x-1)\), we shade the area above the curve to represent all the \(y\) values that satisfy the inequality. It's essential to include the curve itself, indicated by a solid line, because the inequality is 'greater than or equal to'. This visual representation helps us see all possible solutions to the inequality.

The Domain of Logarithmic Functions

The domain of a logarithmic function like \(y = \text{log}_b(x)\) consists of all the possible values of \(x\) that make the function defined and real. Logarithmic functions are only defined for positive numbers, which means their domain is \((0, \text{infinity})\). This restriction comes from the properties of logarithms, which are the inverses of exponential functions, and exponential functions yield positive results for all real numbers. Hence, for our specific function \(y = \text{log}_3(x-1)\), the domain is \(x > 1\), as \(x-1\) must be positive. When graphing, it's crucial to recognize that the graph will approach the vertical line \(x=1\) but never touch it, a characteristic known as a 'vertical asymptote'. Understanding the domain is essential for correctly setting up the graph and ensuring that the solutions to any inequalities fall within the allowable range of values.