Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log ∛(x/y)

Verified step by step guidance

1. Recognize that the expression involves a logarithm of a cube root, which can be rewritten using fractional exponents. Recall that \( \sqrt[3]{a} = a^{1/3} \). So rewrite the expression as \( \log \left( \left( \frac{x}{y} \right)^{1/3} \right) \).
2. Use the logarithm power rule, which states \( \log(a^b) = b \log(a) \), to bring the fractional exponent \( \frac{1}{3} \) in front of the logarithm. This gives \( \frac{1}{3} \log \left( \frac{x}{y} \right) \).
3. Apply the logarithm quotient rule, which states \( \log \left( \frac{a}{b} \right) = \log a - \log b \), to expand \( \log \left( \frac{x}{y} \right) \) into \( \log x - \log y \).
4. Substitute this back into the expression to get \( \frac{1}{3} ( \log x - \log y ) \).
5. Distribute the \( \frac{1}{3} \) across the terms inside the parentheses to write the fully expanded form as \( \frac{1}{3} \log x - \frac{1}{3} \log y \).

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 by expanding or condensing them. For example, log(a/b) = log(a) - log(b) and log(a^n) = n·log(a). These properties are essential for simplifying and expanding logarithmic expressions.

Radicals and Exponents

Radicals can be expressed as fractional exponents, where the nth root of a number is the same as raising it to the power of 1/n. For example, the cube root of x is x^(1/3). Converting radicals to exponents helps apply logarithm power rules effectively when expanding expressions.

Logarithmic Expression Expansion

Expanding logarithmic expressions involves breaking down complex logs into sums, differences, or multiples of simpler logs using the properties of logarithms. This process simplifies expressions and can sometimes allow evaluation without a calculator by recognizing standard log values or simplifying terms.