Problem 71 - Algebra for College Students

Question

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{y}{z}\right) $$

Step-by-Step Solution

Answer

\( \log_b \left( \frac{y}{z} \right) = \log_b y - \log_b z \)

Step 1: Apply the Quotient Rule for Logarithms

The exercise requires expressing \( \log_b \left( \frac{y}{z} \right) \) as a simpler logarithmic expression. Start by applying the quotient rule for logarithms: \[ \log_b \left( \frac{A}{B} \right) = \log_b A - \log_b B \] In this case, set \( A = y \) and \( B = z \).

Step 2: Substitute into the Quotient Rule

Substitute the values into the quotient rule: \[ \log_b \left( \frac{y}{z} \right) = \log_b y - \log_b z \] This simplifies the expression into a difference of two logarithms.

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a fundamental concept. It helps transform a logarithm of a division into a simpler form. This rule states that the logarithm of a quotient is equal to the difference of the logarithms. It is expressed as:

• \( \log_b \left( \frac{A}{B} \right) = \log_b A - \log_b B \)

When using this rule, it is important to ensure almost cheerful accuracy in substituting values for \(A\) and \(B\). For instance, in the expression \(\log_b (\frac{y}{z})\), the values \(A = y\) and \(B = z\) are substituted to get \(\log_b y - \log_b z\).
Remember, the base \(b\) must stay constant for both logarithms. This stability makes computing the solution more straightforward and limits confusion.
Applying the quotient rule can simplify complex expressions, making them easier to manage while solving mathematical problems.

Properties of Logarithms

Understanding properties of logarithms is essential when manipulating logarithmic expressions. These properties allow us to simplify expressions and solve equations efficiently. Here are some of the core properties:

• Product Rule: \( \log_b (AC) = \log_b A + \log_b C \)
• Quotient Rule: \( \log_b \left( \frac{A}{B} \right) = \log_b A - \log_b B \)
• Power Rule: \( \log_b (A^n) = n \cdot \log_b A \)

These properties allow you to rework expressions into addition, subtraction, or multiplication forms, which are often simpler to compute or further manipulate.
Familiarity with these rules helps in tasks like expanding logarithmic expressions or compressing them into compounded or composite forms. Once you master these, tackling complex logarithms becomes far less intimidating.

Simplifying Logarithms

Simplifying logarithms involves breaking down complex expressions into simpler terms using various logarithmic rules. When given expressions like \( \log_b (\frac{y}{z}) \), you can employ the quotient rule to reframe it as \( \log_b y - \log_b z \).
Simplification doesn't stop at the quotient rule. If logarithms involve exponents, you might use the power rule to further simplify. Suppose you have \( \log_b (y^a) \), you can turn this into \( a \cdot \log_b y \) using the power rule.
Striving for a simpler form helps when working with longer calculations or solving equations. It reduces room for error and presents logarithmic expressions clearly. The key is consistent application of these rules, ensuring each step simplifies the expression while maintaining its original equivalence.