Problem 102 - College Algebra and Calculus: An Applied Approach

Question

Condense the expression to the logarithm of a single quantity.\(\ln (x+1)+2 \ln (x-1)+3 \ln x\)

Step-by-Step Solution

Answer

The expression condenses to \(\ln(x^8 - 2x^7 + x^6)\)

Step 1 - Apply the power rule

Using the power rule \(\ln(a^n) = n\ln(a)\), the given expression becomes \(\ln(x+1)+ \ln((x-1)^2) + \ln(x^3)\)

Step 2 - Apply the product rule

Using the product rule \(\ln(ab) = \ln(a) + \ln(b)\), these terms can be condensed into a single logarithm as \(\ln((x+1) \cdot (x-1)^2 \cdot x^3)\)

Step 3 - Simplify the argument

Simplify the expression beneath the log to get \(\ln((x^3) \cdot (x^2 - 2x + 1) \cdot x^3)\) which further simplifies to \(\ln(x^6 \cdot (x^2 - 2x+1)) = \ln(x^8 - 2x^7 + x^6)\)

Key Concepts

Power Rule in Logarithms

The power rule is an essential tool when working with logarithms, especially when needing to simplify expressions. It states that for any positive real number \( a \) and real number \( n \), the logarithm of \( a^n \) is equivalent to multiplying \( n \) by the logarithm of \( a \). In formula terms:

• \( \ln(a^n) = n \cdot \ln(a) \)

This rule allows us to "bring down" exponents in a logarithmic expression, making calculations much more straightforward.
For instance, if you have an expression like \( 2 \ln(x-1) \), using the power rule converts this to \( \ln((x-1)^2) \). This step is crucial when you want to condense or simplify logarithmic expressions. It effectively transforms the multiplication involved with powers into an addition in logs, which can be easier to manage in further calculations.

Product Rule in Logarithms

The product rule for logarithms is another significant property used when condensing expressions. It provides a way to combine multiple logarithms into a single term. The rule states that the logarithm of a product is equal to the sum of the logarithms of its factors:

• \( \ln(a \cdot b) = \ln(a) + \ln(b) \)

This property is particularly helpful when dealing with an expression composed of several logarithmic parts.
Consider an expression \( \ln(x+1) + \ln((x-1)^2) + \ln(x^3) \). By the product rule, you can combine these into a single logarithm: \( \ln((x+1) \cdot (x-1)^2 \cdot x^3) \).
Using the product rule simplifies expressions and facilitates easier calculations in later steps. It transforms multiple terms into one, reducing complexity and making the expression easier to read and interpret.

Logarithm Properties

Logarithm properties are fundamental rules that help simplify and manipulate expressions involving logs. Mastery of these properties is crucial since it involves multiple rules like power rules, product rules, and quotient rules.

• The power rule allows us to manage exponents efficiently: \( \ln(a^n) = n \cdot \ln(a) \).
• The product rule combines multiple log terms: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
• The quotient rule, which isn't directly used here, allows division to be handled: \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \).

These rules allow us to break down complex arguments into more manageable parts in calculations.
For example, in the given exercise, we applied the power and product rules to consolidate various logarithmic terms into one. Finally, this leads us to simplify further using basic algebraic techniques to achieve a concise form: \( \ln(x^8 - 2x^7 + x^6) \).
Meeting the challenge of these exercises involves not only understanding each property but knowing when to use them to manipulate expressions effectively. This knowledge paves the way for solving real-world problems where such expressions appear.