Problem 76 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The expression \(\ln \frac{x y}{z}\) can be expanded into \(\ln(x) + \ln(y) - \ln(z)\) using the properties of logarithms.

Step 1: Identify the Rule Needed

In the exercise, \(\ln \frac{x y}{z}\) is given; it has a quotient inside the logarithm. Therefore, the rule of logarithm of a quotient needs to be used, which states \(\ln(a / b) = \ln(a) - \ln(b)\)

Step 2: Apply Logarithmic Rule on the Expression

Applying the rule of quotient on the expression \(\ln \frac{x y}{z}\) it decomposes into \(\ln(x y) - \ln(z)\)

Step 3: Identify and Apply Another Logarithmic Rule

The expression \(\ln(x y)\) is the logarithm of a product. Therefore, the product rule can be used on it, which states \(\ln(a \cdot b) = \ln(a) + \ln(b)\). Applying the product rule we have: \(\ln(x y) = \ln(x) + \ln(y)\)

Step 4: Substitute The Expression

Substitute the expression \(\ln(x y)\) from step 3 into the equation found in step 2: \(\ln(x y) - \ln(z) = (\ln(x) + \ln(y)) - \ln(z)\)

Step 5: Simplify the Expression

After the substitution the expression simplifies to: \(\ln(x) + \ln(y) - \ln(z)\)

Key Concepts

Logarithmic Rules

Understanding logarithmic rules is essential for solving different mathematical problems that involve logarithms. A logarithm, at its core, is the inverse operation to exponentiation. This means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.

Logarithms have certain rules that help simplify complex expressions and solve equations. The three primary rules are:

• The Product Rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\), which states that the logarithm of a product is the sum of the logarithms of the factors.
• The Quotient Rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\), which is used when dividing two numbers inside a logarithm.
• The Power Rule: \(\log_b(m^n) = n\cdot\log_b(m)\), which applies when a logarithm contains a power.

These rules can drastically simplify logarithmic expressions, making it easier for one to perform calculations or solve equations. Besides these three, there are several other logarithmic identities that can be applied in different scenarios.

Expanding Logarithmic Expressions

Expanding logarithmic expressions is a crucial skill in algebra and calculus. It involves rewriting a single logarithmic expression into a sequence of simpler log terms that are added or subtracted. This process is based on implementing the logarithmic rules.

For example, the expression \(\ln\left(\frac{x y}{z}\right)\) can be expanded using the product and quotient rules. To break it down:

• Firstly, the quotient rule is used to separate the logarithm of a division into the difference of two logarithms.
• Next, if there is a product within a logarithm, as is the case with \(\ln(x y)\), the product rule would be applied to further expand the expression.

The expansion process turns the original logarithmic expression into a more manageable form, especially when solving complex equations. The expanded form also allows for easier differentiation or integration in calculus.

Quotient Rule of Logarithms

The quotient rule of logarithms is a nifty tool in mathematics, essential for breaking down expressions that involve the division of two quantities inside a logarithm. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In its general form, it is expressed as \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\).

When you apply this rule, as seen in our original exercise, you can simplify \(\ln\left(\frac{x y}{z}\right)\) to \(\ln(x y) - \ln(z)\). It's a straightforward rule that, when remembered and applied correctly, simplifies the process of handling division within logarithmic functions. This is particularly beneficial when you are dealing with expressions that will lead to future calculations or transformations.

Product Rule of Logarithms

The product rule of logarithms is equally crucial in maneuvering through the labyrinth of logarithmic expressions. This rule makes multiplying variables within a logarithm a much simpler task. It states that the logarithm of a product is the sum of the logarithms of the individual factors, formally written as \(\log_b(mn) = \log_b(m) + \log_b(n)\).

Using the product rule allows you to expand \(\ln(x y)\) into \(\ln(x) + \ln(y)\). It is particularly useful in cases where you have multiple terms multiplied within a logarithm, as it can break them down into individual log terms that are more manageable. When dealing with complex expressions or when integrating or differentiating logarithmic functions, applying the product rule can greatly simplify your work.