Problem 16 - College Algebra

Question

For the following exercises, condense to a single logarithm if possible. $$ \ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right) $$

Step-by-Step Solution

Answer

-2 ln a + 4 ln b - 5 ln c.

Step 1: Apply Logarithmic Property for Division

According to the property of logarithms \( \ln \left( \frac{x}{y} \right) = \ln x - \ln y \), we can rewrite the given expression: \[ \ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right) = \ln(a^{-2}) - \ln(b^{-4} c^{5}). \]

Step 2: Apply Logarithmic Property for Multiplication

Using the property \( \ln(xy) = \ln x + \ln y \), rewrite the second part of the expression: \[ \ln(b^{-4} c^{5}) = \ln(b^{-4}) + \ln(c^{5}). \] Substituting this back gives: \[ \ln(a^{-2}) - (\ln(b^{-4}) + \ln(c^{5})). \]

Step 3: Simplify Using Logarithmic Power Rule

Apply the logarithmic power rule, \( \ln(x^n) = n \ln x \), to each term: \[ \ln(a^{-2}) = -2 \ln a, \] \[ \ln(b^{-4}) = -4 \ln b, \] and \[ \ln(c^{5}) = 5 \ln c. \] Substitute these into the expression: \[ -2 \ln a - (-4 \ln b + 5 \ln c). \]

Step 4: Simplify the Expression

Distribute the negative sign and simplify: \[ -2 \ln a + 4 \ln b - 5 \ln c. \] This is the expression condensed into a single logarithmic form.

Key Concepts

Logarithmic Properties

Logarithmic properties are essential for simplifying and understanding logarithmic expressions. One fundamental property is the division rule: when you encounter a logarithm of a fraction, like \( \ln \left( \frac{x}{y} \right) \), it can be split into two separate logarithms: \( \ln x - \ln y \). This property is similar to how exponents work in division and allows you to break down complex expressions into simpler, manageable parts.
Logarithmic properties aren't limited to addition and subtraction; they also handle multiplication through the property \( \ln(xy) = \ln x + \ln y \). These basic operations provide a toolkit for transforming and simplifying logarithms, making them easier to work with in different mathematical contexts. They help to isolate variables when solving equations and often convert multiplication or division operations inside logarithms to plain addition or subtraction, which simplifies calculations.

Logarithmic Power Rule

The logarithmic power rule is another powerful tool in the field of logarithms, simplifying expressions that involve powers. It states that for any positive number \( x \) and real number \( n \), \( \ln(x^n) = n \ln x \). This rule allows you to take exponents out of the logarithmic expression, converting the operation into multiplication.
In situations where you have an expression like \( \ln(a^{-2}) \), you can apply this rule to get \( -2 \ln a \). This operation makes calculations more straightforward as it translates complex powers into simple multipliers, which are easier to manage and combine with other terms.
The logarithmic power rule is especially useful when expressions contain multiple terms raised to a power since each term can be individually simplified using this rule. This methodology dramatically simplifies the process of condensing logarithmic expressions into a single term.

Logarithmic Division Rule

The logarithmic division rule is a cornerstone property used in condensing expressions involving logarithms. When you have an expression like \( \ln \left( \frac{x}{y} \right) \), this rule lets you decompose it into \( \ln x - \ln y \). The minus sign signifies the division between the terms \( x \) and \( y \) inside the logarithm.

The division rule is particularly helpful because it transforms a division operation happening inside a logarithm into a subtraction operation outside of it. This change is advantageous when working to simplify expressions because subtraction is typically easier to handle than division, especially when coupled with other logarithmic rules.

• Applying the division rule allows you to separate complex quotient terms into individual components.
• It aids in isolating different parts of an expression, simplifying further manipulation and combination with other logarithmic terms.

Mastery of this rule can significantly reduce the complexity of expressions, enabling more straightforward solutions and deeper insights into mathematical behaviors involving logarithms.

Combining Logarithms

Combining logarithms effectively is a skill that builds on the foundational properties of logarithms to merge separate logarithmic expressions into a single, cohesive form. The process requires careful application of the previously mentioned properties: division, power, and multiplication rules. For instance, expression management like turning \( \ln(x) + \ln(y) \) into \( \ln(xy) \) demonstrates how addition in logarithmic form corresponds to multiplication within.
When tasked with condensing expressions, strategically employ:

• The multiplication property \( \ln(xy) = \ln x + \ln y \), which helps when bringing separate terms under a single logarithm.
• The subtraction form derived from the division rule for handling terms that are subtractions \( \ln x - \ln y = \ln \left( \frac{x}{y} \right) \).

The goal is to leverage these properties to rewrite large, unwieldy logarithmic expressions into neatly condensed forms. This skill is vital for solving equations and simplifying expressions efficiently in mathematics.