Problem 94 - College Algebra and Calculus: An Applied Approach

Question

Condense the expression to the logarithm of a single quantity.\(2 \ln 8+5 \ln z\)

Step-by-Step Solution

Answer

The logarithmic expression simplifies to \( \ln(64z^5) \)

Step 1: Identify logarithmic properties that can be used

In this particular problem, two properties of logarithms are essential: \n 1. The rule of exponentiation: \( a \ln b = \ln (b^a) \) \n 2. The multiplication factor rule: \( \ln m + \ln n = \ln (mn) \)

Step 2: Apply exponentiation rule

Using the rule of exponentiation, we can refactor the expression. So, \(2 \ln 8\) can be refactored to \( \ln (8^2) \) and likewise \(5 \ln z = \ln (z^5) \). So, the logarithmic expression turns into \( \ln (8^2) + \ln (z^5) \)

Step 3: Apply multiplication factor rule

Using the multiplication factor rule, the expression can be further simplified to \( \ln [(8^2) * (z^5)] \)

Step 4: Simplify the expression

Simplify the expression to get the final result. So the final expression becomes \( \ln (64z^5) \)

Key Concepts

Logarithmic Properties

Logarithmic expressions often appear complex, but we can simplify them using fundamental properties. Two crucial properties help us in this: the exponentiation rule and the multiplication factor rule. With these tools, we can take expressions that may seem daunting and break them down into manageable parts.

• Logarithm of a power: When you have something like \(a \ln b\), it translates to \(\ln(b^a)\). This is commonly referred to as the exponentiation rule. It helps transform a multiplied coefficient outside into an exponent inside the logarithm.
• Logarithm of a product: If you see \(\ln m + \ln n\), it becomes \(\ln(mn)\). This is known as the multiplication factor rule. It allows us to combine two separate logarithms into one.

Using these properties, you can consolidate and rewrite logarithmic expressions into more compact forms. The aim of utilizing these rules is to combine terms logically, making your calculations more straightforward and clean.

Exponentiation Rule

The exponentiation rule in logarithms allows us to move a multiplier in front of a logarithm into the exponent of the base itself. This nifty property can transform an expression to make it simpler and reveal a clearer relationship between its components.
When you see a term like \(2 \ln 8\), use the exponentiation rule: \(2 \ln 8 = \ln(8^2)\). Essentially, you're taking the "2" in front and moving it as the power of "8." The same applies to other similar expressions.

• For \(b\ln x\), it turns into \(\ln(x^b)\).
• This rule helps reduce the clutter of the expression, making it easier for other rules and properties to be applied afterwards.

When applying this rule, adjust each term separately before performing any additional steps. This precise handling ensures the expression is condensed as much as possible before it's further manipulated.

Multiplication Factor Rule

The multiplication factor rule, often fondly termed the product rule in logarithms, is a fundamental backbone for combining logarithmic expressions. It allows us to express a sum of logs as a single log of a product. This ability to condense is an essential skill when dealing with complex logarithmic scenarios.
If you have two separate logarithms added together, such as \(\ln u + \ln v\), by using the multiplication factor rule, this becomes \(\ln(uv)\). It's like telling the logarithms to work together to form one unified quantity.

• For expressions like \(\ln m + \ln n\), transform it into \(\ln(mn)\).
• This rule is key when reducing large, separated terms into a single logarithmic expression.

Remember, the multiplication factor rule can only be applied when the logs are added. Always verify that your logs are properly formatted before using this useful property to streamline the expression.