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 [(10x²∛(1 - x))/(7(x + 1)²)]

Verified step by step guidance

1. Identify the logarithmic expression: $\log \left(\frac{10x^2\sqrt[1-x]{3}}{7{x+1}^2}\right)$.
2. Apply the logarithm property for division: $\backslash log\backslash left(\backslash frac\{A\}\{B\}\backslash right)\; =\; \backslash log\; A\; -\; \backslash log\; B$. So rewrite as $\backslash log\backslash left(10x^\{2\}\backslash sqrt[3]\{1-x\}\backslash right)\; -\; \backslash log\backslash left(7(x+1)^\{2\}\backslash right)$.
3. Use the logarithm property for multiplication: $\backslash log(AB)\; =\; \backslash log\; A\; +\; \backslash log\; B$. Expand both logarithms: $\backslash log\; 10\; +\; \backslash log\; x^\{2\}\; +\; \backslash log\; \backslash sqrt[3]\{1-x\}\; -\; \backslash left(\backslash log\; 7\; +\; \backslash log\; (x+1)^\{2\}\backslash right)$.
4. Apply the power rule for logarithms: $\backslash log\; a^\{b\}\; =\; b\; \backslash log\; a$. Rewrite the terms with exponents: $\backslash log\; 10\; +\; 2\; \backslash log\; x\; +\; \backslash frac\{1\}\{3\}\; \backslash log\; (1-x)\; -\; \backslash log\; 7\; -\; 2\; \backslash log\; (x+1)$.
5. Combine all terms into a single expanded expression: $\backslash log\; 10\; +\; 2\; \backslash log\; x\; +\; \backslash frac\{1\}\{3\}\; \backslash log\; (1-x)\; -\; \backslash log\; 7\; -\; 2\; \backslash log\; (x+1)$. Note that $\backslash log\; 10$ can be evaluated if the base is 10.

Key Concepts

Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log(a * b) = log a + log b, log(a / b) = log a - log b, and log(a^n) = n log a. These rules are essential for breaking down complex expressions into simpler terms.

Radicals and Exponents

Understanding how to rewrite radicals as fractional exponents is crucial. For instance, the cube root of (1 - x) can be expressed as (1 - x)^(1/3). This allows the use of the power rule of logarithms to expand terms involving roots effectively.

Simplifying Logarithmic Expressions

After applying logarithmic properties, simplifying the resulting expression by combining like terms or evaluating constants (like log 10) helps in obtaining the most expanded and simplified form. This step often involves recognizing when terms can be evaluated without a calculator.