Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the expression as the logarithm of a single quantity. \(\ln x-5 \ln (x+3)\)

Step-by-Step Solution

Answer

The expression as a single logarithm is: \( \ln (x/ (x+3)^5) \).

Step 1: Conversion using the power rule

First, apply the power rule of logarithms to the term \(5 \ln (x+3)\). This becomes \( \ln ((x+3)^5) \). So, the expression becomes \( \ln x - \ln((x+3)^5) \)

Step 2: Conversion using the quotient rule

Next, apply the quotient rule of logarithms, i.e., \( \ln(a) - \ln(b) = \ln (a/b) \), to the expression obtained in Step 1. This gives the expression as a single logarithm: \( \ln (x/ (x+3)^5) \).

Key Concepts

Logarithms Quotient Rule

Understanding how to simplify logarithmic expressions is crucial when dealing with complex equations. The logarithms quotient rule is a powerful tool for this purpose. In essence, this rule states that the difference between two logarithms with the same base can be expressed as the logarithm of the quotient of their arguments. Formally, if you have two logarithms, \( \log_b(a) - \log_b(c) \), their difference can be rewritten as \( \log_b(\frac{a}{c}) \).

Applying this rule turns a seemingly complex subtraction of two separate logarithmic terms into a single, more manageable expression. This is exactly what we do in the step-by-step solution, where we merge \( \ln x \) and \( -\ln((x+3)^5) \) to result in \( \ln(\frac{x}{(x+3)^5}) \). This change not only simplifies the expression but also sets the stage for further computation or analysis, such as solving for x or evaluating the logarithmic value.

Logarithms Power Rule

Another cornerstone of working with logarithms is the logarithms power rule. Understanding this rule allows you to manipulate and simplify expressions involving logarithms where the argument is raised to a power. The power rule states that if you have an expression of the form \( \log_b(a^k) \), this can be written as \( k \cdot \log_b(a) \).

In the solution to our exercise, we used the power rule to rewrite \(5 \ln (x+3)\) as \(\ln ((x+3)^5)\). The key to this transformation is recognizing that multiplying a logarithm by a coefficient is equivalent to raising its argument to the power of that coefficient. By performing this step, the expression’s complexity is reduced, bringing us closer to representing it as a single logarithmic quantity. It's a powerful simplification technique that often precedes the use of other rules, like the quotient rule.

Expressing Logarithms as a Single Quantity

The ultimate goal in many logarithmic operations is to express logarithms as a single quantity. This concept is especially important because it often precedes the solving phase of an equation and simplifies the expression for further analysis. When logs are combined into a single term, they are more easily manipulated, and it often reveals connections and solutions that are not immediately obvious.

To achieve this, we use the power of the previously mentioned rules. After applying the power rule and the quotient rule sequentially, as demonstrated in our original exercise, we effectively converted the expression \( \ln x - 5 \ln (x+3) \) into a solitary logarithmic term, \( \ln(\frac{x}{(x+3)^5}) \). This condensation of information makes it easier to grasp the relationship within the equation and typically simplifies the process of finding the value of the variable involved. By mastering the use of these properties, algebraic expressions with logarithms become much less daunting and more manageable in mathematical analysis or even in real-world applications.