In Exercises 1–40, 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. Recall the logarithmic property that allows you to bring the exponent in front of the logarithm: $\log b_{}{x}^3=\; 3\log b_{}x$.
2. Apply this property to the given expression $\log b_{}{x}^3$ to rewrite it as $3\log b_{}x$.
3. Since the expression is now expanded, check if $\log b_{}x$ can be simplified further or evaluated directly. Without specific values for $b$ and $x$, it remains as is.
4. Conclude that the fully expanded form of $\log b_{}{x}^3$ is $3\log b_{}x$.
5. Remember that this property is useful for simplifying logarithmic expressions and solving logarithmic equations.

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 Expansion

Logarithmic expansion involves rewriting a logarithm of a product, quotient, or power into a sum, difference, or multiple of logarithms. This process helps simplify complex expressions and solve equations by breaking them into manageable parts, such as expanding log_b(x^3) into 3 * log_b(x).

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator often requires recognizing special values or using properties to simplify expressions. For instance, if x or b are powers of the same base, the logarithm can be evaluated exactly. Understanding these techniques helps in solving problems efficiently without computational tools.