Problem 83 - Precalculus with Limits

Question

Comparing Logarithmic Quantities In Exercises 83 and 84 , compare the logarithmic quantities. If two are equal, then explain why. $$ \frac{\log _{2} 32}{\log _{2} 4}, \quad \log _{2} \frac{32}{4}, \quad \log _{2} 32-\log _{2} 4 $$

Step-by-Step Solution

Answer

The first expression simplifies to \(\frac{5}{2}\), while both the second and third expressions simplify to 3; hence the second and third expressions are equal and the first is different.

Step 1: Simplify the First Expression

The first expression is \(\frac{\log _{2} 32}{\log _{2} 4}\). Since 32 is \(2^5\) and 4 is \(2^2\), we can rewrite the expression as \(\frac{\log _{2} 2^5}{\log _{2} 2^2}\). According to the properties of logarithms, the exponent can be taken out in front of the log, resulting in \(\frac{5\log _{2} 2}{2\log _{2} 2}\). As the logarithm of a number with the very number as base equals one, we obtain \(\frac{5}{2}\).

Step 2: Simplify the Second Expression

The second term is \(\log _{2} \frac{32}{4}\). Again, 32 is \(2^5\) and 4 is \(2^2\), so the expression simplifies to \(\log _{2} \frac{2^5}{2^2}\), which simplifies further to \(\log _{2} 2^3\) because you subtract the denominator's exponent from the numerator's when dividing exponentials of the same base. According to the properties of logarithms again, the 3 comes outside the log, hence it simplifies to 3.

Step 3: Simplify the Third Expression

The third term is \(\log _{2} 32 - \log _{2} 4\). Applying the rules of logarithms, when subtracting one log from another with the same base, you divide the arguments. Hence, this simplifies to \(\log _{2} \frac{32}{4}\), which as per the arguments above, simplifies down to 3.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that simplify expressions and solve equations in various mathematical contexts. One key property is the ability to move exponents in and out of the logarithmic function. For instance, if we have an expression like \(\log_b(a^c)\), it can be rewritten as \(c\log_b(a)\). This is incredibly useful because it gives us a more manageable form to work with.

Another crucial rule of logarithms is the change of base formula, which allows us to switch between different logarithmic bases. However, more often in exercises like the one given, you'll find two properties particularly useful:

• The Product Property: \(\log_b(mn) = \log_b(m) + \log_b(n)\) simplifies the multiplication inside the logarithm.
• The Quotient Property: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\) simplifies the division inside the logarithm.

These properties simplify the way we handle logarithms, making them a potent tool in algebraic manipulations.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions requires the use of logarithmic properties for a step-by-step breakdown of complex forms. Consider the expression \(\frac{\log_2 32}{\log_2 4}\). To simplify, first express the arguments as powers of 2: 32 is \(2^5\) and 4 is \(2^2\). So, we rewrite the expression as \(\frac{\log_2 (2^5)}{\log_2 (2^2)}\).

Utilizing the properties of logarithms, bring down the exponents: \(\frac{5\log_2 2}{2\log_2 2}\). Notice that \(\log_2 2 = 1\) because the base and argument are the same. This reduces our expression to \(\frac{5}{2}\).

Understanding how to deconstruct these expressions systematically is essential in mastering logarithms. The same logarithmic properties help simplify expressions like \(\log_2 \frac{32}{4}\) into \(\log_2 8\), which simplifies further when applying the properties of logarithms.

Logarithm Rules

Logarithm rules are at the heart of understanding and working with logarithms effectively. Each expression benefits from these principles, ensuring consistent and efficient problem-solving.

Key rules include:

• Power Rule: \(\log_b(a^c) = c\log_b(a)\)
• Product Rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• Quotient Rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)

Understanding these three core rules allows us to handle not just simple logarithmic operations but complex calculations by removing ambiguity in expressions. For instance, in the exercise given, the third expression \(\log_2 32 - \log_2 4\) directly applies the Quotient Rule to achieve a straightforward answer. This ability to transition between expressions seamlessly is what mastering logarithm rules grants you.