Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log \left( \frac{10x^2 \sqrt[3]{1 - x}}{7(x + 1)^2} \right)$

Verified step by step guidance

1. Start by recognizing that the logarithm of a quotient can be expressed as the difference of logarithms: \(\log \left( \frac{A}{B} \right) = \log A - \log B\).
2. Apply this property to the given expression: \(\log \left( \frac{10x^{2} \sqrt[3]{1 - x}}{7(x + 1)^{2}} \right) = \log \left( 10x^{2} \sqrt[3]{1 - x} \right) - \log \left( 7(x + 1)^{2} \right)\).
3. Next, use the product property of logarithms: \(\log (AB) = \log A + \log B\), to expand both logarithms: \(\log 10 + \log x^{2} + \log \sqrt[3]{1 - x} - \left( \log 7 + \log (x + 1)^{2} \right)\).
4. Rewrite the logarithms of powers using the power property: \(\log (a^{b}) = b \log a\). So, \(\log x^{2} = 2 \log x\), \(\log \sqrt[3]{1 - x} = \log (1 - x)^{1/3} = \frac{1}{3} \log (1 - x)\), and \(\log (x + 1)^{2} = 2 \log (x + 1)\).
5. Combine all parts to write the fully expanded expression: \(\log 10 + 2 \log x + \frac{1}{3} \log (1 - x) - \log 7 - 2 \log (x + 1)\).

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(ab) = log a + log b, log(a/b) = log a - log b, and log(a^n) = n log a. These rules help break down complex expressions into simpler sums and differences of logs.

Radicals and Exponents

Radicals such as cube roots can be expressed as fractional exponents, e.g., ∛(1 - x) = (1 - x)^(1/3). Understanding how to rewrite radicals as exponents allows the use of logarithm power rules to simplify expressions involving roots and powers.

Evaluating Logarithms Without a Calculator

Some logarithmic values can be simplified or evaluated exactly using known log values and properties, especially when the arguments are products or powers of numbers like 10. Recognizing these can help simplify expressions without a calculator, such as log(10) = 1.