Problem 74 - 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. $$\log _{2} \frac{x^{5} y^{3}}{3}$$

Step-by-Step Solution

Answer

\( 5 \cdot \log_2 x + 3 \cdot \log_2 y - \log_2 3 \)

Step 1: Apply the Quotient Property of Logarithms

The quotient property of logarithms states that \( \log_b \frac{M}{N} = \log_b M - \log_b N \). Using this, rewrite the expression as \( \log_2 (x^5 y^3) - \log_2 (3) \).

Step 2: Apply the Product Property of Logarithms

The product property of logarithms is \( \log_b (MN) = \log_b M + \log_b N \). Apply this property to \( \log_2 (x^5 y^3) \) to get \( \log_2 (x^5) + \log_2 (y^3) \).

Step 3: Apply the Power Property of Logarithms

The power property of logarithms states that \( \log_b (M^n) = n \cdot \log_b M \). First apply this to \( \log_2 (x^5) \) to get \( 5 \cdot \log_2 x \). Then apply it to \( \log_2 (y^3) \) to get \( 3 \cdot \log_2 y \).

Step 4: Combine All Parts

Substitute back into the expression from Step 1: \( \log_2 \frac{x^5 y^3}{3} = 5 \cdot \log_2 x + 3 \cdot \log_2 y - \log_2 3 \).

Key Concepts

Quotient Property of Logarithms

The quotient property of logarithms is a fundamental tool in simplifying logarithmic expressions. This property states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Specifically, for any base \( b \), and positive numbers \( M \) and \( N \), the property is expressed as: \[ \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \] This property can help break down complex expressions into simpler parts.

• It allows subtraction of two logarithms when dealing with a division inside the log.
• This is particularly useful for expressions with variables, making them easier to manage and solve.

In our example, \( \log_2 \left( \frac{x^5 y^3}{3} \right) \), the logarithm becomes: \[ \log_2 (x^5 y^3) - \log_2 (3) \] By reducing the problem into two separate logarithms, each term can be addressed individually, making the expression easier to manipulate and understand.

Product Property of Logarithms

The product property of logarithms is another key rule that simplifies expressions where multiplication appears inside the logarithm. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. If \( b \) is the base and \( M \) and \( N \) are positive numbers, then: \[ \log_b (M \times N) = \log_b M + \log_b N \] Using this property helps separate the components of a multiplication inside a log into smaller, manageable parts:

• This simplifies the expression, turning a product into a sum of logs.
• It is especially handy for expressions involving products of variables or multi-term expressions.

In our example, consider the expression \( \log_2 (x^5 y^3) \). By applying the product property, it can be expanded to: \[ \log_2 (x^5) + \log_2 (y^3) \] This transformation makes it easier to apply additional properties, like the power property, since each part of the product is now resolved independently.

Power Property of Logarithms

Finally, the power property of logarithms deals with expressions that involve exponents within a logarithm. According to this property, the logarithm of a number raised to an exponent can be rewritten as the exponent multiplied by the logarithm of the base number. Formally, for any positive number \( M \), exponent \( n \), and base \( b \), the property is given by: \[ \log_b (M^n) = n \cdot \log_b M \] This property is extremely useful for simplifying expressions with powers:

• It eliminates the exponent by converting it into a coefficient.
• Using this property can greatly simplify equations and expressions involving large exponents.

In the current exercise, the power property is applied to both terms derived from the product property:

• \( \log_2 (x^5) \) becomes \( 5 \cdot \log_2 x \).
• \( \log_2 (y^3) \) simplifies to \( 3 \cdot \log_2 y \).

By converting powers to coefficients, the expression \( \log_2 (x^5 y^3) \) becomes much simpler, ultimately making mathematical operations and simplifications more straightforward.