Problem 59 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 4 \ln (x+6)-3 \ln x $$

Step-by-Step Solution

Answer

\[ \ln \frac {(x+6)^4}{x^3} \]

Step 1: Apply the Power Rule for Logarithms

By using the power rule, the expression can be rewritten as: \( \ln (x+6)^4 - \ln x^3 \)

Step 2: Apply the Quotient Rule for Logarithms

Using the quotient rule, simplify the expression into a single logarithm: \( \ln \frac {(x+6)^4}{x^3} \)

Step 3: Simplify the Expression

The resultant expression is already a single logarithm with a coefficient of 1, as required. Therefore, no further simplification is needed

Key Concepts

Properties of Logarithms

Logarithms, much like exponents, have unique properties that simplify complex expressions. These properties allow us to break down or combine logarithmic terms. Understanding these properties can greatly simplify calculations that use logs. The main properties of logarithms include the Product Rule, the Quotient Rule, and the Power Rule.

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Rule: This is applicable when you're dealing with the logarithm of a division or quotient: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• Power Rule: This simplifies a logarithm with an exponent: \( \log_b (M^n) = n \cdot \log_b M \).

These properties are fundamental in manipulating and simplifying logarithmic expressions, especially when transforming them into a format that is more manageable for calculations. Each property builds upon the concept of logs as the inverse of exponentiation.

Power Rule for Logarithms

The Power Rule is particularly helpful when dealing with logarithmic expressions that involve exponents. It allows the exponent to "come down" and multiply the logarithm, making the expression easier to manage. This rule is typically used in the initial steps of simplifying a logarithmic expression.For example, in the expression \( 4 \ln (x+6) \), we can apply the Power Rule. Here's how it works:1. You see that the number "4" in front of the logarithm can be turned into an exponent: \( \ln (x+6)^4 \).2. This transformation makes it easier to further simplify the expression.The Power Rule is simple but powerful because it converts a multiplication problem into a power one, making subsequent steps like using the Quotient Rule more straightforward. Understanding and applying the Power Rule correctly is crucial when simplifying expressions involving multiple logarithmic components.

Quotient Rule for Logarithms

The Quotient Rule provides an efficient way to handle the division of numbers within a logarithm. It is hugely beneficial for condensing logarithmic expressions into a single log term. This is particularly useful when you have expressions that involve subtractions of logs.When applying the Quotient Rule, keep in mind:

• The rule states that \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• This means you can take two logarithmic expressions with a subtraction operation and condense them into one logarithm representing the division of their arguments.

In the exercise, after using the Power Rule to rewrite \( \ln (x+6)^4 \) and \( \ln x^3 \), you can see how the Quotient Rule simplifies the expression:1. Start with \( \ln (x+6)^4 - \ln x^3 \).2. Apply the Quotient Rule to get \( \ln \frac{(x+6)^4}{x^3} \).The Quotient Rule effectively consolidates terms, allowing you to achieve an expression as a single logarithm. This makes it easier to evaluate or further manipulate, particularly when dealing with complex or multiple logarithmic terms.