Problem 21 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{x y}{z} $$

Step-by-Step Solution

Answer

\(\ln x + \ln y - \ln z\)

Step 1: Understand Logarithms Properties

There are specific properties of logarithms that will be used in this particular problem. Multiplication inside a logarithm is the same as addition outside of the logarithm, and of course, division inside a logarithm is the same as subtraction outside of the logarithm. Hence, \(\ln \frac{a}{b} = \ln a - \ln b \) and \(\ln ab = \ln a + \ln b\).

Step 2: Apply Logarithm Properties to Divide the Fraction

The given expression is \(\ln \frac{xy}{z}\). By using the division property of logarithms, this expression can be rewritten as \(\ln xy - \ln z\).

Step 3: Apply Logarithm Properties to Multiply the Terms

The term \(\ln xy\) can be broken down further using the property of multiplication inside a logarithm. By applying this property, \(\ln xy\) becomes \(\ln x + \ln y\).

Step 4: Combine the Results

The full expansion of the original expression is then given by combining the results found in step 2 and 3. This gives us \(\ln x + \ln y - \ln z\) as the expanded form of the original logarithmic expression.

Key Concepts

Logarithm Expansion

Logarithm expansion is all about breaking down complex logarithmic expressions into simpler components. Imagine you’re unboxing a big present and breaking it down into smaller, easier-to-manage pieces.
You can think of logarithm expansion as following the rulebook that tells us how to unpack each part step by step. This method is especially helpful when dealing with products and quotients inside a logarithm, just like in the given example.

In the expression \( \ln \frac{xy}{z} \), the goal is to expand it by applying specific properties of logarithms. This involves turning the expression into something easier to process or more workable for further calculations.

• The division part \( \ln \frac{xy}{z} \) turns into \( \ln xy - \ln z \).
• Then, the multiplication part within the logarithm \( \ln xy \) can be expanded into \( \ln x + \ln y \).

Using these steps helps simplify complex expressions and makes further algebraic procedures more manageable.

Natural Logarithm

A natural logarithm is a type of logarithm that uses the constant \( e \) (approximately 2.718) as its base. If you hear "natural log," it's referring to logs with this special base.
The symbol \( \ln \) is used to represent natural logarithms. While other bases, like 10 (common logarithms) or 2, can be handy depending on the context, natural logarithms are particularly useful in calculus and many areas of science and engineering.
The natural logarithm possesses the same basic properties as logarithms of other bases, except that all these properties revolve around the number \( e \). This means we can expand, simplify, and manipulate them using the same principles.
In the example given, the use of \( \ln \) indicates we’re working completely in the realm of natural logarithms. Each part of our logarithm expansion is executed under the base \( e \), though we typically don’t need to represent \( e \) explicitly in solving these basic expansions, as it remains consistent throughout. Understanding and recognizing when the natural logarithm is being used is key to handling expressions involving \( \ln \).

Multiplication and Division Properties of Logarithms

Logarithms have some pretty cool properties involving multiplication and division, which help us break down complex expressions. These properties make solving and manipulating logarithmic expressions a lot easier.
1. **Multiplication Property:** If you have multiplication inside a logarithm, it translates to addition outside. Hence, \( \ln(ab) = \ln a + \ln b \). In our problem, this helps expand \( \ln(xy) \) into \( \ln x + \ln y \).
2. **Division Property:** With division, logarithms turn the operation into subtraction. So, \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \). In this example, initially, \( \ln\left(\frac{xy}{z}\right) \) becomes \( \ln(xy) - \ln z \).

These properties give you the power to unravel a logarithm's complex expressions into simpler parts. Using them, you can resolve expressions step-by-step until you reach a straightforward answer. They’re invaluable tools in everything from solving equations to proving mathematical theorems in advanced areas. By getting comfortable with these properties, you gain a strong foundational skill in math.