Problem 511 - Precalculus

Question

Rewrite \(\ln (7 r \cdot 11 s t)\) in expanded form.

Step-by-Step Solution

Answer

The expanded form is \( \ln(7) + \ln(r) + \ln(11) + \ln(s) + \ln(t) \).

Step 1: Identify the Product Inside the Logarithm

The expression inside the logarithm is a product: \(7r \cdot 11st\). Notice that \(7\), \(r\), \(11\), \(s\) and \(t\) are multiplied together.

Step 2: Apply the Logarithmic Product Rule

Use the logarithmic product rule, which states that \( \ln(ab) = \ln(a) + \ln(b) \). Apply this rule to break down the product inside the logarithm.

Step 3: Break Down the Expression Inside the Log

Apply the product rule to expand \( \ln(7r \cdot 11st) \) as follows: \( \ln(7) + \ln(r) + \ln(11) + \ln(s) + \ln(t) \).

Step 4: Final Expanded Form

The expanded form of \( \ln(7r \cdot 11st) \) is \( \ln(7) + \ln(r) + \ln(11) + \ln(s) + \ln(t) \). This is the expression rewritten with all components separated by addition.

Key Concepts

Logarithmic Product Rule

The logarithmic product rule is a fundamental concept in logarithms. It allows us to express the logarithm of a product as a sum of individual logarithms. The rule is represented by the formula:

• \( \ln(ab) = \ln(a) + \ln(b) \)

This rule simplifies the process of handling complex logarithmic expressions, especially when dealing with multiplication of terms. For example, if you have a product inside a natural logarithm like \( \ln(7r \cdot 11st) \), you can break it down using the product rule.
Using the rule, the expression expands as follows:

• First, identify elements in the product: \(7, r, 11, s, t\).
• Apply the rule to separate these elements: \( \ln(7) + \ln(r) + \ln(11) + \ln(s) + \ln(t) \).

This simplification helps in solving equations and analyzing expressions in a more manageable form.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a specific type of logarithm. Its base is the mathematical constant \(e \approx 2.71828\), known as Euler's number.
The natural logarithm is commonly used in calculus, physics, and many other branches of science and engineering due to its unique properties.
Some important features of the natural logarithm include:

• Inverse Function: The natural logarithm is the inverse of the exponential function \(e^x\).
• Derivative: The derivative of \(\ln(x)\) is \(1/x\), which makes it useful in calculus for integration and differentiation tasks.
• Exponential Growth: Natural logarithms are used to model phenomena that exhibit exponential growth or decay.

In the example \( \ln(7r \cdot 11st) \), we used the properties of the natural logarithm to simplify the expression via the product rule.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations such as addition, subtraction, multiplication, and division. In logarithmic contexts, particularly when dealing with expressions inside a logarithm, understanding algebraic expressions is crucial.
For example, in the expression \(7r \cdot 11st\), understanding:

• Constants: Numerical values such as \(7\) and \(11\).
• Variables: Symbols like \(r, s,\) and \(t\), which can represent unknowns or varying quantities.

This understanding helps in applying rules like the logarithmic product rule successfully.
When breaking down an algebraic expression, recognize the different parts and how they interact in products to use logarithmic rules effectively. In the example provided, each component of the product is separated into its own section in the expanded logarithmic form, demonstrating the clarity that expansion can provide.