Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression as one logarithm. $$ 2 \ln x-4 \ln (1 / y)-3 \ln (x y) $$

Step-by-Step Solution

Answer

\( \ln \left( \frac{y}{x} \right) \) represents the expression as one logarithm.

Step 1: Apply Logarithm Power Rule

The logarithm power rule states that \( a \ln b = \ln b^a \). Using this rule, rewrite each term:\[ 2 \ln x = \ln(x^2) \]\[ 4 \ln \left( \frac{1}{y} \right) = \ln\left(\left(\frac{1}{y}\right)^4\right) = \ln\left(\frac{1}{y^4}\right) \]\[ 3 \ln(x y) = \ln((x y)^3) \]

Step 2: Combine the Logarithms Using Properties

Use the properties of logarithms to combine the expressions into a single logarithm. Start by recalling the properties:1. \( \ln a + \ln b = \ln(ab) \)2. \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \)Apply these properties:\[ \ln(x^2) - \ln\left(\frac{1}{y^4}\right) - \ln((x y)^3) \]Reuse property 2 to get:\[ = \ln\left(\frac{x^2}{1/y^4}\right) - \ln((x y)^3) \]Simplifying within, it continues as:\[ = \ln(x^2 y^4) - \ln((x y)^3) \] (multiplying with the negative sign tackled by division next)

Step 3: Simplify Further

Now, apply the properties again:\[ \ln(x^2 y^4) - \ln((x y)^3) = \ln\left(\frac{x^2 y^4}{(x y)^3}\right) \]

Step 4: Final Simplification

Simplify the inside of the logarithm: first, expand and simplify:\[ \frac{x^2 y^4}{(x y)^3} = \frac{x^2 y^4}{x^3 y^3} = \frac{y}{x} \]Thus the expression becomes:\[ \ln\left(\frac{y}{x}\right) \]

Key Concepts

Logarithm Properties

Logarithms are incredibly helpful in simplifying complex expressions. They have properties that make it easier to merge multiple logarithmic terms. For instance, one key property is the **power rule**. This rule states that multiplying a logarithm by a number is equivalent to raising the base to that power and then taking the logarithm.

• Power Rule: \( a \ln b = \ln(b^a) \)

In the given expression, this rule transforms each term, such as \( 2 \ln x \) to \( \ln(x^2) \). Another important property is the **product rule**:

• Product Rule: \( \ln a + \ln b = \ln(ab) \)

This allows us to combine addition of logs into a log of a product. There is also the **quotient rule**:

• Quotient Rule: \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \)

Understanding these properties is crucial as they facilitate the transition from multiple logarithmic terms into a single, consolidated expression.

Logarithmic Simplification

The goal of simplifying logarithmic expressions is often to express them in the simplest form possible, sometimes reducing multiple terms to a single logarithm. To do this, applying the properties of logarithms is essential.
In our specific case, we begin by applying the power rule to each term in the given expression. For example, \( 3 \ln(x y) \) converts to \( \ln((x y)^3) \). This approach helps handle exponents systematically.

• Simplifying \( 4 \ln \left( \frac{1}{y} \right) \) follows the same logic: first transform it to \( \ln\left(\left(\frac{1}{y}\right)^4\right) = \ln\left(\frac{1}{y^4}\right) \).

After dealing with exponents, combining logs using the quotient rule is the next step. This involves subtracting logs, which is equivalent to dividing their arguments. Eventually, it results in consolidating all terms into a single logarithm.

Algebraic Manipulation

In mathematics, especially in logarithmic expressions, performing algebraic manipulation is fundamental to achieving a simplified result.
For instance, once each term has been transformed using logarithmic rules, combining them requires thoughtful use of algebra. Initially, we rewrote the expression using the power rule.
Then, we carefully applied the quotient rule to group the terms together.

• The expression \( \ln(x^2) - \ln\left(\frac{1}{y^4}\right) - \ln((x y)^3) \) becomes \( \ln\left(\frac{x^2 y^4}{(x y)^3}\right) \).

Finally, simplifying the fraction \( \frac{x^2 y^4}{x^3 y^3} \) results in \( \frac{y}{x} \). This requires careful division of the terms to ensure all components are simplified correctly. Such steps demonstrate the power of algebraic manipulation in deriving a more direct and neat representation of complex expressions.