Problem 47 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\ln \frac{e x y}{z}\)

Step-by-Step Solution

Answer

\(1 + \ln(x) + \ln(y) - \ln(z)\)

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule for logarithms states that \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). Apply the rule to the given expression \(\ln\frac{exy}{z}\), resulting in \(\ln(exy) - \ln(z)\).

Step 2: Apply the Product Rule of Logarithms to the Numerator

The product rule for logarithms states that \(\ln(ab) = \ln(a) + \ln(b)\). Apply this rule to \(\ln(exy)\), which gives \(\ln(e) + \ln(x) + \ln(y)\).

Step 3: Combine and Simplify the Expression

Combine the results from Steps 1 and 2: \(\ln(e) + \ln(x) + \ln(y) - \ln(z)\). Since \(\ln(e) = 1\), the expression simplifies to \(1 + \ln(x) + \ln(y) - \ln(z)\).

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a powerful tool that helps you break down expressions involving division inside a logarithm. When you encounter a logarithm of a fraction, such as \( \ln\left(\frac{a}{b}\right) \), you can separate it into two simpler logarithms. This is done by writing it as \( \ln(a) - \ln(b) \). This rule is particularly useful because it transforms a division operation into a subtraction, simplifying the expression and making it easier to handle.

In our original exercise, we used the quotient rule to address the fraction \( \ln\frac{exy}{z} \). By applying the rule, we were able to simplify it into \( \ln(exy) - \ln(z) \). This simplification is the first key step when dealing with complex logarithmic expressions.

Understanding this rule enables you to carefully dissect expressions, making it much simpler to solve algebraic equations or simplify other similar problems.

Product Rule for Logarithms

The product rule for logarithms is equally handy, especially when dealing with multiplication inside a logarithmic expression. According to this rule, for any two quantities \( a \) and \( b \), the rule can be expressed as \( \ln(ab) = \ln(a) + \ln(b) \). This formula allows us to turn multiplication into addition, which is often easier to work with algebraically.

In the context of our example, after applying the quotient rule, we ended up with \( \ln(exy) \) in the numerator. Using the product rule here, we can further simplify it to \( \ln(e) + \ln(x) + \ln(y) \).

By mastering the product rule, you can effortlessly handle and simplify more complex logarithmic terms, transforming them into a series of simpler additions, leading to a cleaner and clearer solution.

Logarithmic Simplification

Logarithmic simplification is all about using the various logarithmic rules to rewrite a complex expression in a more manageable form. This often involves applying both the quotient and product rules, as well as other logarithmic identities, to break down and "tidy up" expressions.

In our problem, after employing both the quotient and product rules, we arrived at the expression \( \ln(e) + \ln(x) + \ln(y) - \ln(z) \). A crucial realization during the simplification stage was recognizing that \( \ln(e) \) equals 1 because \( e \) is the base of natural logarithms. This enables us to further simplify the expression to \( 1 + \ln(x) + \ln(y) - \ln(z) \).

The goal of logarithmic simplification is to rewrite expressions in the simplest form possible. Simplifying logarithms not only makes calculations easier but also reduces the chance of error and makes the solution more aesthetically pleasing. Understanding this process is crucial for tackling complex mathematical problems efficiently.