Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log₃ [9 (x+2) ]

Verified step by step guidance

1. Recognize that the function is given as \(f(x) = \log_{3} \left[ 9 (x+2) \right]\). The goal is to use logarithm properties to rewrite this expression in a simpler form.
2. Recall the logarithm product property: \(\log_{a} (MN) = \log_{a} M + \log_{a} N\). Apply this to separate the logarithm of the product inside the argument: \(f(x) = \log_{3} 9 + \log_{3} (x+2)\).
3. Evaluate \(\log_{3} 9\) by expressing 9 as a power of 3. Since \$9 = 3^2$, use the power property of logarithms: \(\log_{3} 9 = \log_{3} (3^2) = 2\).
4. Substitute this value back into the expression to get \(f(x) = 2 + \log_{3} (x+2)\), which is a simpler form of the original function.
5. To graph \(f(x)\), start with the graph of \(y = \log_{3} (x+2)\), which is a logarithmic function shifted 2 units to the left, then shift the entire graph up by 2 units to account for the constant term.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. For example, the product rule states that log_b(MN) = log_b(M) + log_b(N). These properties allow us to rewrite complex logarithmic expressions into simpler forms, which is essential for understanding and graphing the function.

Change of Base and Logarithmic Functions

A logarithmic function log_b(x) is the inverse of the exponential function b^x. Understanding the base and how it affects the graph's shape and growth is crucial. The base 3 in this problem means the function grows slowly compared to base 10 or e, influencing the graph's steepness.

Graphing Logarithmic Functions

Graphing logarithmic functions involves identifying key features such as domain, vertical asymptotes, intercepts, and transformations. For ƒ(x) = log_3[9(x+2)], rewriting the function helps reveal shifts and stretches, making it easier to plot points and sketch the graph accurately.