Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln x y z\)

Short answer

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

Step-by-step solution

Identify the terms inside the logarithm

The initial expression is \(\ln xyz\). The terms inside the logarithm are x, y and z. These are being multiplied together.

Apply logarithmic properties

According to the multiplication property, the logarithm of a product of variables equals the sum of the logarithms of these variables.

Execute the operation

Apply this property to the initial expression: \(\ln xyz\) expands into \(\ln x + \ln y + \ln z\).

Key concepts

Logarithmic Properties

Logarithms are a fundamental concept in mathematics and have distinctive properties that allow us to manipulate and simplify complex expressions. A key logarithmic property is that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors.

Another important property is the logarithm of a quotient, which states that the logarithm of a division between two numbers is the difference of the logarithms of the numerator and the denominator. Lastly, the power property of logarithms indicates that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.

These properties greatly help in solving algebraic problems involving logarithms, as they can transform multiplicative relationships into additive ones, making the expressions much easier to work with.

Expanding Logarithms

Expanding logarithms is a technique used to break down complex logarithmic expressions into simpler forms using logarithmic properties. For example, the expression \(\log_b(mn)\) can be expanded into \(\log_b(m) + \log_b(n)\) because of the logarithm of product rule.

Similarly, expanding a logarithm of a power, such as \(\log_b(m^n)\), would result in \(n \cdot \log_b(m)\). These transformations are particularly useful when solving equations involving logarithms, as they simplify the terms and help in isolating the variable of interest.

An understanding of how to properly expand logarithms can be a powerful tool for students, aiding them not only in solving equations but also in calculus, where logarithmic differentiation is involved.

Logarithms of Product

The logarithm of a product is one of the most commonly used logarithmic properties. It dictates that the logarithm of a product is equal to the sum of the logarithms of each of the factors. Mathematically, this is expressed as \(\log_b(mn) = \log_b(m) + \log_b(n)\).

This rule simplifies calculations and is particularly helpful when the product involves variables, as it allows each variable to be handled individually. For instance, our initial problem \(\ln(xyz)\) can be expanded using this rule into \(\ln(x) + \ln(y) + \ln(z)\), where \(x\), \(y\), and \(z\) are variables representing positive numbers.

It’s crucial for students to remember that this property only applies to multiplication within the logarithm and that all parts must be multiplied together for this property to be used.

Natural Logarithms

The natural logarithm, denoted as \(\ln\), is a specific type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. Natural logarithms are especially important in the fields of calculus and differential equations because the derivative of \(\ln(x)\) is \(1/x\), making them integral in the process of integration and differentiation.

In our expansion example \(\ln(xyz)\), we used the natural logarithm to express a product of variables. Understanding how to work with natural logarithms is essential for students embarking on advanced mathematics courses.

Whether calculating compound interest in financial contexts or solving for time in half-life radioactive decay problems, the natural logarithm is frequently encountered across various real-world applications.