Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log₂ (96) - log₂ (3)

Verified step by step guidance

1. Identify the logarithmic expression given: \(\log_{2}(96) - \log_{2}(3)\).
2. Recall the logarithmic property for subtraction: \(\log_{a}(x) - \log_{a}(y) = \log_{a}\left(\frac{x}{y}\right)\), where \(a\), \(x\), and \(y\) are positive and \(a \neq 1\).
3. Apply this property to combine the expression into a single logarithm: \(\log_{2}\left(\frac{96}{3}\right)\).
4. Simplify the fraction inside the logarithm: calculate \(\frac{96}{3}\) to get a simpler argument for the logarithm.
5. Rewrite the expression as \(\log_{2}(\text{simplified value})\), which is the condensed form with coefficient 1.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the quotient rule states that log_b(A) - log_b(B) = log_b(A/B), which is essential for condensing expressions.

Logarithmic Expression Simplification

Simplifying logarithmic expressions involves rewriting them as a single logarithm with coefficient 1. This often requires applying properties of logarithms and factoring numbers inside the log to simplify or evaluate without a calculator.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves expressing numbers as powers of the base or factoring to simplify. For example, recognizing that 96/3 = 32, and since 32 = 2^5, log_2(32) = 5, allows exact evaluation.