Problem 67 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{6 x}{y}$$

Step-by-Step Solution

Answer

\( \log_2 \frac{6x}{y} = \log_2 6 + \log_2 x - \log_2 y \)

Step 1: Apply the Quotient Property

The quotient property of logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). Apply this property: \( \log_2 \frac{6x}{y} = \log_2 (6x) - \log_2 (y) \).

Step 2: Apply the Product Property

The product property of logarithms states that \( \log_b (MN) = \log_b M + \log_b N \). Apply this to \( \log_2 (6x) \): \( \log_2 (6x) = \log_2 6 + \log_2 x \).

Step 3: Rewrite the Expression

Substitute the expression from Step 2 back into the equation from Step 1: \( \log_2 \frac{6x}{y} = (\log_2 6 + \log_2 x) - \log_2 y \).

Step 4: Simplify the Expression

Simplify the final expression: \( \log_2 \frac{6x}{y} = \log_2 6 + \log_2 x - \log_2 y \).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that allow us to rewrite expressions in more manageable forms. These properties are tools that simplify the manipulation of logarithmbased equations. Here are some key properties:

• Product Property: The logarithm of a product is the sum of the logarithms of the individual factors. For example, \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms. For example, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base. For example, \( \log_b (M^n) = n \cdot \log_b M \).

Understanding these properties helps streamline solving logarithmic expressions. They are essential for simplifying complex mathematical expressions, making calculations easier.

Quotient Property

The quotient property of logarithms makes it simpler to handle divisions inside a logarithmic expression. When you encounter a logarithm of a fraction, you can split it into two separate terms. Consider this formula: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This translates division into subtraction, which is generally easier to compute or further transform.
In our exercise, \( \log_2 \frac{6x}{y} \) is transformed into \( \log_2 (6x) - \log_2 y \) using the quotient property. This step breaks down the initial logarithm into smaller parts that can be more easily simplified or calculated further.

Product Property

The product property of logarithms is handy when dealing with multiplication inside a logarithm. It allows you to decompose a logarithm of a product into the sum of separate logarithms.Here's the formula: \( \log_b (MN) = \log_b M + \log_b N \).
In our given exercise, we further broke down the log expression \( \log_2 (6x) \) to \( \log_2 6 + \log_2 x \) using the product property.

• This lets us deal with simpler, individual logarithms, potentially making it easier to compute or combine with other terms.
• It’s a useful property whenever you encounter a multiplicative factor in a logarithmic expression.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves applying properties of logarithms in a strategic manner to reduce complexity. Here, we make an expression cleaner and easier to work with.
In the original exercise, we arrived at the equation: \( \log_2 \frac{6x}{y} = \log_2 6 + \log_2 x - \log_2 y \). This chained the steps:

• Applying quotient property to separate division into subtraction.
• Using the product property to handle multiplication by addition.

Each of these moves contributes to a clearer, linear representation of a logarithmic equation.
The simplified expression allows for easier calculations, comparisons, or further transformations if necessary. Using these properties consistently can reveal insights into the mathematical relationships within logarithmic equations.