In Exercises 1–40, 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: $\log {\mathrm{\_}}_4\left(\frac{64}{y}\right)=\log {\mathrm{\_}}_{4}64-\log {\mathrm{\_}}_{4}y$.
2. Next, express 64 as a power of 4, since the base of the logarithm is 4. Note that 64 = 4^3 because 4^3 = 4 × 4 × 4 = 64.
3. Use the logarithmic power rule: $\log {\mathrm{\_}}_4{4}^3=\; 3\; \times \log {\mathrm{\_}}_{4}4$.
4. Since $\log {\mathrm{\_}}_{4}4=\; 1$, simplify the expression to just 3.
5. Combine all parts to write the expanded form: $3\; -\log {\mathrm{\_}}_{4}y$.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to expand or condense logarithmic expressions. For example, the quotient rule states that log_b(M/N) = log_b(M) - log_b(N), which is essential for expanding log4(64/y).

Change of Base and Evaluating Logarithms

Evaluating logarithms without a calculator often involves expressing numbers as powers of the base. For instance, 64 can be written as 4^3 since 4^3 = 64. This helps simplify log4(64) to 3, making the expression easier to evaluate.

Simplifying Algebraic Expressions

After applying logarithmic properties, simplifying the resulting algebraic expression is crucial. This includes combining like terms and expressing the answer in the simplest form, such as writing log4(64/y) as log4(64) - log4(y).