Problem 48 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\ln \frac{5 p}{e}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\ln 5 + \ln p - 1\).

Step 1: Apply the Quotient Rule for Logarithms

Recall the logarithm property \(\ln \frac{a}{b} = \ln a - \ln b\). This allows us to split the logarithm of a quotient into the difference of two logarithms. Applying this property to \(\ln \frac{5p}{e}\), we get: \[ \ln \frac{5p}{e} = \ln (5p) - \ln (e) \] Since \(e\) is the base of the natural logarithm, \(\ln(e) = 1\).

Step 2: Apply the Product Rule to First Term

Now, we focus on the term \(\ln (5p)\). We can use the product rule of logarithms: \(\ln (ab) = \ln a + \ln b\). Here, consider \(5p = 5 \cdot p\). So, \[ \ln (5p) = \ln 5 + \ln p \] Now we substitute back into our expression from Step 1.

Step 3: Substitute and Simplify

Substitute the expression for \(\ln(5p)\) back into the expression found in Step 1:\[ \ln (5p) - \ln e = (\ln 5 + \ln p) - 1 \]This simplifies to:\[ \ln 5 + \ln p - 1 \].

Step 4: Final Simplified Expression

After simplifying, the expression \(\ln \frac{5p}{e}\) as the sum and/or difference of logarithms becomes: \[ \ln 5 + \ln p - 1 \].

Key Concepts

Quotient Rule for Logarithms

The Quotient Rule for Logarithms is a cornerstone in understanding how to manipulate and simplify expressions involving logarithms. It states that the logarithm of a quotient is the difference of the logarithms. In mathematical form, this rule is written as \( \ln \frac{a}{b} = \ln a - \ln b \). This might sound a bit abstract, but let's break it down. Imagine you're trying to find out how many times one number fits into another. When we use the Quotient Rule, we're essentially separating the operation of division into two simpler log operations—subtraction, which is more intuitive.

• When you see a fraction inside a logarithm, think of it as a subtraction waiting to happen.
• This rule can greatly simplify complex logarithmic expressions and make calculations more straightforward.

In our problem with \( \ln \frac{5p}{e} \), applying the Quotient Rule allows us to break it down into the difference \( \ln (5p) - \ln (e) \). Since \( \ln(e) \) is 1, it simplifies the expression further right away. This simple rule aids in dissecting and simplifying logarithmic expressions effectively.

Product Rule for Logarithms

The Product Rule for Logarithms is another fundamental tool that can ease your way through complex algebraic expressions involving logs. This rule tells us that the logarithm of a product is the sum of the logarithms of the factors, written as \( \ln (ab) = \ln a + \ln b \). Let's explore this further:

• Whenever you see a product inside a logarithm, it signals that you can expand it using addition.
• This expansion helps to simplify or solve equations that might otherwise be stuck.

In our exercise, the use of the Product Rule on \( \ln (5p) \) results in \( \ln 5 + \ln p \). So, you're effectively unpacking a product into more manageable parts. Remember, breaking down these terms into sums of logarithms helps to simplify and often aid in solving equations. The Product Rule is especially useful in situations involving complex or multiple-variable products within logarithmic expressions.

Natural Logarithm

The natural logarithm is one of the most commonly encountered logarithms in various fields, from pure mathematics to practical sciences. Notated as \( \ln \), it is the logarithm to the base of the mathematical constant \( e \), approximately equal to 2.71828. The natural logarithm provides several advantages:

• It appears naturally in many growth processes, such as population dynamics, interest calculations, and decay problems.
• Natural logarithms simplify calculus operations due to their base \( e \), making differentiation and integration more straightforward.

In the context of our original problem, knowing that \( \ln(e) = 1 \) is pivotal. This small fact helps to simplify expressions involving \( e \), as seen when addressing \( \ln \frac{5p}{e} \). This step not only reduced our fraction to a simpler form but illustrates how natural logarithms are integral to simplifying expressions. Understanding the importance and applications of the natural logarithm enhances the comprehension of many mathematical concepts and real-world applications.