Expand: $lo{g}_8\left(\frac{4\surd x}{64y3}\right)$

Verified step by step guidance

1. Rewrite the expression inside the logarithm to make it easier to work with. The expression is \( \frac{4\sqrt{x}}{64y^3} \). Recognize that \(4\sqrt{x} = 4x^{1/2}\) and \(64 = 8^2\).
2. Use the logarithm property for division: \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). So, \( \log_8 \left( \frac{4x^{1/2}}{64y^3} \right) = \log_8(4x^{1/2}) - \log_8(64y^3) \).
3. Apply the logarithm property for multiplication: \( \log_b(MN) = \log_b(M) + \log_b(N) \). Expand both terms: \( \log_8(4) + \log_8(x^{1/2}) - [\log_8(64) + \log_8(y^3)] \).
4. Use the power rule of logarithms: \( \log_b(a^c) = c \log_b(a) \). Rewrite \( \log_8(x^{1/2}) = \frac{1}{2} \log_8(x) \) and \( \log_8(y^3) = 3 \log_8(y) \).
5. Now, the expression is \( \log_8(4) + \frac{1}{2} \log_8(x) - \log_8(64) - 3 \log_8(y) \). This is the fully expanded form using logarithm properties.

Key Concepts

Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow us to simplify complex 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 the given expression into manageable parts.

Change of Base and Simplifying Bases

Understanding how to express numbers as powers of the logarithm's base is crucial. Here, recognizing that 8, 4, and 64 are powers of 2 allows simplification. This helps rewrite the expression inside the log in terms of base 8 or base 2 for easier expansion.

Radicals and Exponents

Converting radicals to fractional exponents is essential for simplifying expressions inside logarithms. For example, the fourth root of x is x^(1/4). This conversion allows the use of the power rule of logarithms to bring exponents outside the log for easier manipulation.