Problem 45 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{2} 96-\log _{2} 3\)

Step-by-Step Solution

Answer

The expression \(\log _{2} 96-\log _{2} 3\) condenses to \(5\).

Step 1: Apply the property of logarithmic subtraction

Applying the property of logarithmic subtraction, \(\log _{2} 96-\log _{2} 3\) can be rewritten as \(\log _{2} (96/3)\)

Step 2: Simplify the expression

Divide 96 by 3 to get \(\log _{2} 32\)

Step 3: Evaluate the logarithmic expression

Finally, evaluate \(\log _{2} 32\) to get 5 because \(2^5 = 32\)

Key Concepts

Logarithmic Subtraction

Logarithmic subtraction is a powerful tool in algebra that helps simplify expressions involving differences of logarithms with the same base. While this process may sound daunting, it's actually based on a straightforward principle. Subtraction of two logarithms with the same base, such as \( \log_b(x) - \log_b(y) \), can be expressed as a single logarithm of the division of their arguments; that is, \( \log_b(\frac{x}{y}) \).

This property mirrors the rules of exponents, where the quotient of powers with the same base can be simplified by subtracting their exponents. Thus, when you see an expression like \( \log_{2} 96 - \log_{2} 3 \), you can condense it into \( \log_{2}(\frac{96}{3}) \), as the single logarithm of a quotient.

Condensing Logarithms

Condensing logarithms refers to the process of combining multiple logarithmic expressions into a single one. This simplification makes the expression easier to work with and can reveal solutions that may not be immediately evident. The rules to condense logarithms stem from the properties of logarithms which are analogous to the rules of exponents.

To condense a subtraction of logarithms, like in our example, we transform the subtraction into division within a single logarithm. This method can also be applied to other operations, such as adding logarithms turning into multiplication. Mastery of condensing logarithms not only simplifies expressions but also fosters a deeper understanding of the relationship between logarithms and exponents.

Evaluating Logarithms

Evaluating logarithms is the process of finding the exponent that the base of the logarithm must be raised to obtain the argument of the logarithm. For instance, with \( \log_2(32) \), we're looking for the power that 2 must be raised to produce 32. The answer is 5 because \( 2^5 = 32 \).

Being able to evaluate logarithms without a calculator requires familiarity with the properties of exponents. When logarithms appear in their simplest form, as in the final step of our example, the evaluation becomes a matter of recognizing patterns in the powers of numbers. This skill not only helps in simplifying expressions but can also aid in solving equations where the variable is the exponent—the expressive domain of logarithmic functions.