In Exercises 41–70, 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. Recall the logarithmic property that states: $\log {\mathrm{a}}_b-\log {\mathrm{a}}_c=\log {\mathrm{a}}_{\frac{b}{c}}$. This means the difference of two logarithms with the same base can be written as the logarithm of a quotient.
2. Identify the base and the arguments in the expression: the base is 2, the first argument is 96, and the second argument is 3.
3. Apply the property by writing the expression as a single logarithm: $log\_2\; \backslash left(\; \backslash frac\{96\}\{3\}\; \backslash right)$.
4. Simplify the fraction inside the logarithm: calculate $\backslash frac\{96\}\{3\}$ to get a simpler argument.
5. Write the final condensed logarithmic expression as $log\_2\; (\backslash text\{simplified\; value\})$, which is a single logarithm 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 Condensation

Condensing logarithmic expressions means rewriting multiple logs as a single logarithm. This involves applying properties to combine terms into one log with coefficient 1, simplifying the expression and making it easier to evaluate or interpret.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing numbers as powers of the base. For example, expressing 96/3 as 32, and knowing 32 = 2^5, helps find log_2(32) = 5 directly, simplifying the evaluation process.