Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log₃ (x+1)/9

Verified step by step guidance

1. Identify the given function: \(f(x) = \log_{3} \left(x + \frac{1}{9}\right)\). The goal is to rewrite it using properties of logarithms to simplify or transform the expression for easier graphing.
2. Recall the logarithm property for sums inside the log: \(\log_b (a + c)\) cannot be directly separated into simpler log terms. However, check if the argument \(x + \frac{1}{9}\) can be rewritten as a product or quotient to apply log properties like \(\log_b (mn) = \log_b m + \log_b n\) or \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\).
3. Since \(x + \frac{1}{9}\) is a sum, try to express it in a form involving multiplication or division. For example, consider if \(x + \frac{1}{9}\) can be rewritten as \(\frac{9x + 1}{9}\), which is a quotient.
4. Use the logarithm quotient property: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\). Applying this to \(\log_3 \left(\frac{9x + 1}{9}\right)\) gives \(\log_3 (9x + 1) - \log_3 9\).
5. Simplify \(\log_3 9\) since \$9 = 3^2\(, so \)\log_3 9 = 2\(. Thus, the function can be rewritten as \)f(x) = \log_3 (9x + 1) - 2$. This form is easier to analyze and graph.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules that simplify logarithmic expressions. For example, log_b(xy) = log_b(x) + log_b(y) and log_b(x^k) = k log_b(x). These properties help rewrite complex logarithmic functions into simpler forms for easier analysis and graphing.

Logarithmic Function Transformation

Transformations of logarithmic functions involve shifts, stretches, and reflections that change the graph's position or shape. Adding or subtracting constants outside the log shifts the graph vertically, while changes inside the log affect horizontal shifts. Understanding these helps in sketching the graph accurately after rewriting the function.

Graphing Logarithmic Functions

Graphing logarithmic functions requires knowledge of their domain, range, intercepts, and asymptotes. The basic log function has a vertical asymptote where the argument is zero and passes through (1,0). After rewriting the function using properties, these features guide the sketching of the graph.