Retaining the Concepts. Expand: log7 (5√x/49y¹⁰) fifth root of x

Verified step by step guidance

1. Rewrite the expression clearly: \( \log_7 \left( \frac{5 \sqrt{x}}{49 y^{10}} \right) \times \sqrt[5]{x} \). Note that \( \sqrt{x} = x^{1/2} \) and \( \sqrt[5]{x} = x^{1/5} \).
2. Express all radicals as exponents: \( \log_7 \left( \frac{5 x^{1/2}}{49 y^{10}} \right) \times x^{1/5} \).
3. Use the logarithm property for division inside the log: \( \log_7 (5 x^{1/2}) - \log_7 (49 y^{10}) \).
4. Apply the logarithm product rule to the numerator and denominator terms: \( \log_7 5 + \log_7 x^{1/2} - (\log_7 49 + \log_7 y^{10}) \).
5. Use the power rule for logarithms: \( \log_7 5 + \frac{1}{2} \log_7 x - \log_7 49 - 10 \log_7 y \), then multiply the entire expression by \( x^{1/5} \) as indicated.

Key Concepts

Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow us to expand or simplify logarithmic expressions. For example, log_b(M/N) = log_b(M) - log_b(N), and log_b(M^k) = k * log_b(M). These rules help break down complex log expressions into simpler parts.

Radicals and Exponents

Radicals like roots can be rewritten as fractional exponents, e.g., the fifth root of x is x^(1/5). Understanding this conversion is essential for manipulating expressions inside logarithms and applying logarithmic power rules effectively.

Simplifying Algebraic Expressions

Simplifying expressions inside the logarithm, such as combining constants and variables with exponents, is crucial before expansion. This includes rewriting terms like 49 as 7^2 and expressing roots and powers consistently to facilitate the use of logarithmic properties.