Problem 48 - Intermediate Algebra

Question

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 4. $$ \ln \frac{5 p}{e} $$

Step-by-Step Solution

Answer

\( \ln(5p) - \ln(e) = \ln(5) + \ln(p) - 1 \)

Step 1: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that the logarithm of a quotient is the difference of the logarithms. Applying this to \( \ln \frac{5p}{e} \), we get: \( \ln \frac{5p}{e} = \ln(5p) - \ln(e) \).

Step 2: Simplify Using Properties of Natural Logarithms

We know that \( \ln(e) = 1 \), because the natural logarithm of the constant \( e \) is 1 by definition. Therefore, substitute 1 for \( \ln(e) \): \( \ln(5p) - \ln(e) = \ln(5p) - 1 \).

Step 3: Apply the Product Rule for Logarithms

The product rule states that the logarithm of a product is the sum of the logarithms. Apply this rule to \( \ln(5p) \): \( \ln(5p) = \ln(5) + \ln(p) \).

Step 4: Combine the Resulting Expressions

Now substitute back into the expression from Step 2: \( \ln(5) + \ln(p) - 1 \). This is the expression of the original logarithm written as a sum and/or difference of simpler logarithms.

Key Concepts

Quotient Rule

The Quotient Rule for logarithms is a helpful tool when dealing with a logarithm of a fraction. This rule states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. Think of it as splitting the fraction into two separate, simpler logarithms. For instance, when you apply the quotient rule to \( \ln \frac{5p}{e} \), it transforms into two distinct logarithms, becoming \( \ln(5p) - \ln(e) \). This approach helps break down complex expressions, making them easier to work with and understand.

Product Rule

The Product Rule for logarithms makes dealing with products inside logarithms straightforward. According to this rule, the logarithm of a product is equal to the sum of the logarithms of the factors. This means if you have a multiplication inside a logarithm, you can separate it into two simpler parts. For example, taking the expression \( \ln(5p) \), we use the product rule to split it into \( \ln(5) + \ln(p) \). This breakdown allows for easier calculation and interpretation, simplifying the manipulation of logarithmic expressions.

Natural Logarithm

A natural logarithm is a specific type of logarithm that uses the base \( e \), where \( e \approx 2.718 \.\.\. \), a constant very common in mathematics. The natural logarithm, denoted by \( \ln \), often simplifies calculations in calculus and exponential functions. An essential property to remember is that \( \ln(e) = 1 \). This property plays a significant role when simplifying expressions involving \( e \), as seen when \( \ln(e) \) is replaced with 1. Natural logarithms are intrinsic in connecting complex exponential and algebraic expressions.

Simplifying Expressions

Simplifying expressions using logarithmic laws involves breaking down complex logarithmic forms into simpler parts. For instance, by combining the rules of logs, you can transform a cumbersome term into an expanded form that is more manageable. After applying both the quotient and product rule, and simplifying \( \ln(5p) - \ln(e) \) into \( \ln(5) + \ln(p) - 1 \), the expression is much clearer. This process makes it easier to analyze and solve logarithmic equations, ensuring you can understand the underlying mathematical structure without getting lost in complicated terms.