Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log_b x³

Verified step by step guidance

1. Identify the logarithmic expression given: \(\log_{b} x^{3}\).
2. Recall the logarithmic power rule: \(\log_{b} (a^{c}) = c \cdot \log_{b} a\).
3. Apply the power rule to the expression: \(\log_{b} x^{3} = 3 \cdot \log_{b} x\).
4. Since \(\log_{b} x\) cannot be simplified further without additional information, the expanded form is \$3 \log_{b} x$.
5. Thus, the expression is fully expanded using properties of logarithms.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, the power rule states that log_b(x^n) = n * log_b(x), which is essential for expanding expressions like log_b(x^3).

Logarithmic Functions and Notation

A logarithm log_b(x) answers the question: to what power must the base b be raised to get x? Understanding this definition helps in manipulating and interpreting logarithmic expressions. The base b must be positive and not equal to 1, and x must be positive for the logarithm to be defined.

Evaluating Logarithms Without a Calculator

Some logarithmic expressions can be simplified or evaluated exactly using known values or properties, such as log_b(b) = 1 or log_b(1) = 0. Recognizing when an expression can be rewritten to use these values helps avoid calculator use and deepens understanding of logarithmic behavior.