Problem 4 - College Algebra with Corequisite Support

Question

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln (3 a b \cdot 5 c)\)

Step-by-Step Solution

Answer

\( \ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c) \).

Step 1: Break Down the Logarithmic Expression

The given expression is \( \ln (3ab \cdot 5c) \). Start by rewriting the expression without parentheses: \( \ln (3ab5c) \). This can be expanded using the properties of logarithms.

Step 2: Apply Product Rule of Logarithms

According to the product rule, \( \ln(xy) = \ln(x) + \ln(y) \). In this expression, treat \( 3ab \cdot 5c \) as the product of \( 3ab \) and \( 5c \). Therefore, \( \ln (3ab \cdot 5c) = \ln (3ab) + \ln (5c) \).

Step 3: Further Expand Each Logarithm

Now apply the product rule separately to both parts. For \( \ln(3ab) \), apply the product rule again: \( \ln(3ab) = \ln(3) + \ln(a) + \ln(b) \). Similarly, apply it to \( \ln(5c) \) to obtain: \( \ln(5c) = \ln(5) + \ln(c) \).

Step 4: Combine Everything

Combine all the expressions from Step 3 to get the fully expanded logarithmic expression: \( \ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c) \). This shows the complete expansion of the original logarithm into a sum of simpler logarithms.

Key Concepts

Product Rule of Logarithms

When learning about logarithms, one of the first key things to understand is the product rule. This rule is a fundamental aspect of logarithms and helps you to break down complex logarithmic expressions.
Essentially, the product rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, if you have two numbers, say \(x\) and \(y\), then the rule is expressed as:

• \(\ln(xy) = \ln(x) + \ln(y)\)

This rule is extremely helpful when you're trying to expand expressions like \( \ln(3ab \cdot 5c) \).
By applying the product rule, you can simplify these product expressions step by step, turning them into a series of simpler terms. This makes solving the expression easier and often provides deeper insights into the relationships between the factors involved.

Properties of Logarithms

Logarithms come with a set of powerful properties that make them versatile tools in mathematics. Understanding these properties can make your work with logarithms much easier and efficient.
Here are some of the core properties:

• Product Rule: As discussed, this property helps convert the logarithm of a product into a sum.
• Quotient Rule: Allows you to handle division inside a logarithm, expressed as \(\ln(\frac{x}{y}) = \ln(x) - \ln(y)\).
• Power Rule: Simplifies handling exponents, shown by \(\ln(x^a) = a \cdot \ln(x)\).
• Change of Base: Converts logarithms of different bases, useful in calculations and expressed as \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\).

These properties are not just theoretical concepts; they have practical applications in breaking down complex problems.
For the expression \( \ln(3ab \cdot 5c) \), combining these properties allows you to express it fully in terms of simpler logarithmic terms.

Logarithmic Expression

A logarithmic expression like \( \ln(3ab \cdot 5c) \) may look daunting at first, but breaking it down into parts makes it manageable. The goal with expanding these expressions is to express them as sums, differences, or products of simpler logarithms.
This allows mathematicians and students to engage with difficult-looking expressions by converting them into more familiar and workable forms.

• The given expression, \( \ln(3ab \cdot 5c) \), combines numbers and variables within a single logarithm.
• Through the process of expansion, you can rewrite it step by step using the product rule, resulting in a straightforward expression like \( \ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c) \).

This method isn’t only a solution for an exercise; it’s a key technique in mathematics for managing and simplifying expressions. Once you get the hang of expanding logarithmic expressions, you'll see how much simpler and clearer problems become.