Graph each function. ƒ(x) = log₆ (x-2)

Verified step by step guidance

1. Identify the base of the logarithm and the argument inside the log function. Here, the function is given as \(f(x) = \log_{6}(x - 2)\), where the base is 6 and the argument is \((x - 2)\).
2. Determine the domain of the function by setting the argument greater than zero because the logarithm is only defined for positive arguments. So, solve the inequality \(x - 2 > 0\) to find the domain.
3. Find the vertical asymptote of the graph, which occurs where the argument of the logarithm equals zero. Set \(x - 2 = 0\) and solve for \(x\) to locate the vertical asymptote.
4. Plot key points by choosing values of \(x\) greater than 2, substituting them into the function \(f(x) = \log_{6}(x - 2)\), and calculating the corresponding \(y\) values (without final numeric evaluation here).
5. Sketch the graph using the vertical asymptote as a boundary, the plotted points for shape guidance, and remember that the graph increases slowly to the right since the base 6 is greater than 1.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as f(x) = log_b(x), where b is the base. It answers the question: to what power must the base b be raised to produce x? Understanding the properties of logarithms is essential for graphing and interpreting these functions.

Domain of Logarithmic Functions

The domain of a logarithmic function f(x) = log_b(x - h) consists of all x-values for which the argument (x - h) is positive. For f(x) = log_6(x - 2), the domain is x > 2, meaning the graph only exists to the right of x = 2, which acts as a vertical asymptote.

Graphing Transformations of Logarithmic Functions

Graphing f(x) = log_6(x - 2) involves shifting the basic log_6(x) graph horizontally by 2 units to the right. Recognizing how horizontal shifts affect the position of the graph and its asymptotes helps in accurately sketching the function.