Problem 81 - Algebra for College Students

Question

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ 2 \log _{b} x-4 \log _{b} y $$

Step-by-Step Solution

Answer

\(\log_{b} \left( \frac{x^2}{y^4} \right)\) as a single logarithm.

Step 1: Apply the Power Rule of Logarithms

The power rule of logarithms states that \( a \log_b M = \log_b M^a \). Using this, we can rewrite the expressions: \( 2 \log_b x = \log_b x^2 \) and \(-4 \log_b y = \log_b y^{-4} \).

Step 2: Combine Using the Product Rule

The product rule of logarithms states that \( \log_b A + \log_b B = \log_b (AB) \). For the given expression, we combine the rewritten terms from Step 1: \( \log_b x^2 - \log_b y^4 = \log_b (x^2) + \log_b (y^{-4}) \).

Step 3: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log_b A - \log_b B = \log_b (\frac{A}{B}) \). Therefore, \( \log_b (x^2) - \log_b (y^4) \) can be combined into \( \log_b \left( \frac{x^2}{y^4} \right) \).

Key Concepts

Power Rule

The power rule of logarithms is an essential mathematical principle that helps simplify logarithmic expressions. Imagine you have a logarithmic expression with a coefficient multiplying the log, such as \( a \log_b M \). The power rule allows you to transform this into \( \log_b M^a \). This transformation is handy because it simplifies complex expressions by consolidating powers into the base of the logarithm.

In the given problem, the expressions \( 2 \log_b x \) and \( -4 \log_b y \) are simplified using the power rule. They become \( \log_b x^2 \) and \( \log_b y^{-4} \), respectively.

The process:

• Take the coefficient of the log, such as 2 from \( 2 \log_b x \).
• Use it as an exponent, thus converting to \( \log_b x^2 \).
• Apply the same to other terms, even if they have a negative coefficient, like \(-4 \log_b y \) which becomes \( \log_b y^{-4} \).

Once simplified, these expressions are much easier to manage in further calculations, as they allow for easier combination with other log expressions.

Product Rule

When working with logarithms, combining terms is made much simpler by using the product rule. The product rule states that \( \log_b A + \log_b B = \log_b (AB) \). This means, if you have two logarithmic terms added together, you can simplify them into a single log that combines the two values inside.

In our exercise, after applying the power rule, we have terms \( \log_b x^2 \) and \( \log_b y^{-4} \). Although the original operation between these terms is subtraction, applying the power rule yields a scenario where both terms would be added directly if their signs were positive, such as in \( \log_b x^2 + \log_b y^{-4} \).

This step involves:

• Identifying positive and negative terms after the power adjustment.
• Understanding that the product rule combines them for addition to \( \log_b (x^2 \cdot y^{-4}) \).

Remember, even when negative exponents are involved, as long as we think of them as additions first, we can grasp that the terms are naturally combined with opposite operations, guiding us to curtail complex logs into a single term effectively.

Quotient Rule

The quotient rule assists in simplifying logarithmic expressions especially when the terms are initially subtracted. This principle is articulated as \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \). It highlights how subtraction within logs can be combined into one logarithm involving division.

Returning to our example, after using the product rule, terms show inclusion of inversely positioned log values, essentially the system retains subtraction by formulating \( \log_b x^2 \) minus \( \log_b y^4 \). This naturally leads us to use the quotient rule, resulting in \( \log_b \left( \frac{x^2}{y^4} \right) \).

Breaking it down:

• Locate terms where subtraction exists, like \( \log_b x^2 - \log_b y^{4} \).
• Apply the quotient rule, simplifying the expression to a single log showing division of the quantities as \( \log_b \left( \frac{x^2}{y^4} \right) \).

This rule is particularly useful for students looking to express complex calculations succinctly, offering a simple way to consolidate subtraction by reframing it through division in the logarithmic context.