Problem 69 - 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, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x y z $$

Step-by-Step Solution

Answer

\( \log_b(xyz) = \log_b(x) + \log_b(y) + \log_b(z) \)

Step 1: Identify the Logarithmic Expression

We need to evaluate or simplify: \(\).

Step 2: Apply Logarithm Rules

We use the properties of logarithms: \(\log_b(xy) = \log_b x + \log_b y\), \(\log_b(x/y) = \log_b x - \log_b y\), \(\log_b(x^n) = n\log_b x\).

Step 3: Simplify and Compute

Applying these rules step by step.

Step 4: State the Result

\( \log_b(xyz) = \log_b(x) + \log_b(y) + \log_b(z) \)

Key Concepts

Product Rule

In the world of logarithms, understanding the product rule is essential. The product rule states that the logarithm of a product is the sum of the logarithms of individual factors. Mathematically, this is expressed as \( \log_b(mn) = \log_b(m) + \log_b(n) \). This rule is very handy when dealing with logarithms, as it allows us to break down complex expressions into simpler parts, making them easier to manage and understand.

For instance, in the original exercise, the expression \( \log_b(xyz) \) represents the logarithm of a product of three variables \( x \), \( y \), and \( z \). By employing the product rule, you can split this into \( \log_b(x) + \log_b(y) + \log_b(z) \). Each logarithm now represents a simpler form, making it easier to work with and analyze.

Properties of Logarithms

Logarithms have several important properties that help simplify expressions. Knowing these properties enables students to tackle more complex problems. Some key properties include:

• Product Rule: As discussed, \( \log_b(mn) = \log_b(m) + \log_b(n) \). This rule simplifies products inside a logarithm into sums outside the logarithm.
• Quotient Rule: This property states that the logarithm of a quotient is the difference between the logs of the numerator and the denominator: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).


• Power Rule: This property allows you to move exponents in and out of the logarithm: \( \log_b(m^n) = n\log_b(m) \).

By using these properties, complex logarithmic equations can be broken down into simpler, more manageable parts. These properties interconnect and allow for the simplification of mathematical expressions.

Simplification of Logarithms

Simplifying logarithms involves using the properties and rules of logarithms to break down complex logs into more manageable parts. The goal is to express the logarithmic statement in its simplest form, often as sums or differences of smaller logarithms.

For example, with the expression \( \log_b(xyz) \), applying the product rule simplifies it to \( \log_b(x) + \log_b(y) + \log_b(z) \). This process of simplification helps in making complicated logarithmic expressions easier to interpret and solve.

Other rules, such as the quotient and power rules, may also be employed during simplification. For instance, turning \( \log_b(x^3) \) into \( 3\log_b(x) \), or \( \log_b\left(\frac{x}{y}\right) \) into \( \log_b(x) - \log_b(y) \). Understanding and applying these rules reduces the complexity of logarithmic expressions and enhances problem-solving efficiency.