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 [(10x^2∛(1 - x))/(7(x + 1)^2)]

Key Concepts

Properties of Logarithms

The properties of logarithms include rules such as the product rule, quotient rule, and power rule. The product rule states that the logarithm of a product is the sum of the logarithms of the factors. The quotient rule indicates that the logarithm of a quotient is the difference of the logarithms. The power rule allows us to bring exponents in front of the logarithm as a multiplier. Understanding these properties is essential for expanding logarithmic expressions.

Logarithmic Expansion

Logarithmic expansion involves rewriting a logarithmic expression into a sum or difference of simpler logarithmic terms. This process utilizes the properties of logarithms to break down complex expressions into more manageable parts. For example, the expression log(a/b) can be expanded to log(a) - log(b). Mastery of this technique is crucial for simplifying logarithmic expressions and solving related problems.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves calculating the value of a logarithm based on known values or properties. For instance, log base 10 of 100 is evaluated as 2 because 10^2 = 100. In some cases, logarithmic expressions can be simplified to known values without a calculator, especially when dealing with common bases like 10 or e. This skill is important for quickly solving logarithmic equations and understanding their applications.