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Chapter 1: Problem 55

Use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{x y}{z} $$

Short Answer

Expert verified
The expanded form of the given log expression, \( \ln \frac{x y}{z} \), is \( \ln(x) + \ln(y) - \ln(z)\).

Step by step solution

01

Identify the Division Property

We first notice that the logarithm operates on a division (i.e. \(x y / z\)). The division property of logarithms states that the logarithm of a fraction a/b can be rewritten as the difference of two logarithms, \(\ln(a) - \ln(b)\). So, the given expression \( \ln \frac{x y}{z} \) can be rewritten as \( \ln(x y) - \ln(z)\).
02

Apply the Multiplication Property

The multiplication property of logarithms states that the logarithm of a product of two numbers can be rewritten as the sum of the logarithms of those individual numbers. Looking at the first part of our expression from Step 1, \( \ln(x y)\), which represents the logarithm of a product, can be expanded using this property as \( \ln(x) + \ln(y)\).
03

Combine the results

Now we substitute from step 2 into the expression we got from step 1: \( \ln(x) + \ln(y) - \ln(z)\). And this is the final expanded form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Logarithms
Logarithms have some interesting properties that make simplifying and expanding expressions much easier. These properties help in transforming complex logarithmic expressions into simpler ones.

Some key properties of logarithms include:
  • Product Property: Allows us to express the logarithm of a product as a sum of logarithms.
  • Quotient or Division Property: Converts the logarithm of a quotient into the difference of two logarithms.
  • Power Rule: Transforms the logarithm of a power into the multiplication of a logarithm and an exponent.
These properties are essential tools in algebra and calculus to deal with exponential and logarithmic equations. Understanding and applying them effectively is crucial for simplifying logarithmic expressions.
Division Property of Logarithms
When dealing with the logarithm of a fraction, the division property becomes quite handy. This property states that the logarithm of a fraction can be represented as the difference between two logarithms.

For instance, for a fraction \(\frac{a}{b}\), the logarithm can be expanded as:
  • \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
By utilizing this property, fractions in logarithms are broken down, making it possible to separate terms and simplify expressions.

In our exercise, the expression \(\ln\left(\frac{xy}{z}\right)\) utilizes this property to separate into \(\ln(xy) - \ln(z)\). This is the crucial first step in further expanding the expression.
Multiplication Property of Logarithms
The multiplication property is vital when we have products within a logarithmic expression. This property allows us to turn a logarithm of a product into the sum of logarithms.

According to this property:
  • \(\ln(ab) = \ln(a) + \ln(b)\)
It breaks a complex product into simpler individual components that are easier to handle and solve.

For our exercise, after applying the division property, the expression \(\ln(xy)\) is further broken down using the multiplication property. It becomes \(\ln(x) + \ln(y)\). Thus, the complete expansion of \(\ln\left(\frac{xy}{z}\right)\) leads to \(\ln(x) + \ln(y) - \ln(z)\).

This step-by-step expansion shows how useful the multiplication property is in simplifying logarithmic expressions that involve products.

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