Problem 12 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(\log _{2} 9-\log _{2} 3\)

Step-by-Step Solution

Answer

The simplified single logarithm expression is \( \log _{2} 3 \)

Step 1: Identify the given expression

Here, the given expression is \( \log _{2} 9 - \log _{2} 3 \)

Step 2: Apply the Quotient Rule

One of the properties of logarithms, the Quotient Rule, allows us to subtract logarithms by dividing their arguments. This can be formulated as: \( \log_b a - \log_b c = \log_b (a/c) \). This property applies here as both the logarithms have the same base (2). Using this rule, the original expression can be simplified to \( \log _{2} (9/3) \)

Step 3: Simplify the expression

Evaluate the division inside the logarithm. \( 9 ÷ 3 = 3 \). So, the expression simplifies to \( \log _{2} 3 \)

Key Concepts

Logarithm Properties

Logarithms might seem tricky at first, but they're just a way to express exponents in a different form. To work effectively with logarithms, it's essential to understand the core properties they possess. Let’s list some of the most important ones:

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of its factors. Mathematically, it's expressed as: \( \log_b (ac) = \log_b a + \log_b c \).
• Quotient Rule: Similar to the Product Rule, the Quotient Rule gives us a shortcut for subtracting logarithms, which we'll explore more in-depth later. It's shown as: \( \log_b (a/c) = \log_b a - \log_b c \).
• Power Rule: When you have an exponent within a logarithm, the Power Rule allows you to "bring down" the exponent by turning it into a coefficient: \( \log_b (a^c) = c \cdot \log_b a \).

These properties make manipulating and simplifying logarithms much easier and are the basis for most operations involving logarithmic expressions.
Understanding these properties is key to unlocking the potential simplifications that complex logarithmic expressions may offer.

Quotient Rule

The Quotient Rule is one of the essential logarithmic properties you will often use, especially when simplifying expressions. It allows us to turn subtraction of logarithms into a division inside a single logarithm. Here's how it works in more detail:
When you have two logarithms with the same base that are subtracted, \( \log_b a - \log_b c \), the Quotient Rule permits us to write this as just one logarithm with the quotient of their arguments: \( \log_b (a/c) \).

This rule is convenient because:

• It reduces the complexity of the expression.
• It combines two logarithmic terms into one.
• Works only if both logarithms have the same base.

In our original problem, \( \log_2 9 - \log_2 3 \), we applied the Quotient Rule because the bases of both logarithms were the same (base 2). Hence, it simplifies to \( \log_2 (9/3) \). This transformation is particularly useful because it sets you up perfectly for the next step of simplification.

Simplifying Logarithms

Simplifying logarithmic expressions is all about making them as concise as possible by using the properties of logarithms. Once you've applied a rule, such as the Quotient Rule, the next step is to simplify the argument within the logarithm itself.

In the original exercise, after applying the Quotient Rule, we were left with \( \log_2 (9/3) \). This expression can be simplified further by calculating the division within the parentheses, simplifying down to \( \log_2 3 \).

To simplify logarithms effectively:

• First, ensure all possible logarithm properties have been used to combine terms.
• Next, perform any arithmetic operations possible within the argument of the logarithm.
• Lastly, ensure the final result is as simple as it can be, sometimes writing a single logarithmic term.

By simplifying this way, we can make complex logarithmic expressions much more manageable and easier to interpret.