Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log₄ (64/y)

Verified step by step guidance

1. Recall the logarithmic property for the logarithm of a quotient: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\).
2. Apply this property to the given expression: \(\log_4 \left( \frac{64}{y} \right) = \log_4 64 - \log_4 y\).
3. Recognize that 64 is a power of 4 since \$64 = 4^3$.
4. Use the logarithmic identity \(\log_b (b^k) = k\) to simplify \(\log_4 64\) as \(\log_4 (4^3) = 3\).
5. Write the fully expanded expression as \$3 - \log_4 y$.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow you to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N), which is essential for breaking down complex expressions.

Change of Base and Evaluating Logarithms

Evaluating logarithms often involves expressing numbers as powers of the base. For instance, 64 can be written as 4^3, so log_4(64) = 3. Recognizing these relationships helps simplify expressions without a calculator.

Simplifying Algebraic Expressions

Simplifying expressions like log_4(64/y) requires understanding how to separate terms using logarithm properties and handle variables correctly. This involves rewriting the expression as log_4(64) - log_4(y) and simplifying each part individually.