Problem 83 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(\ln \left(\frac{x}{z}+x\right)-\ln \left(\frac{y}{z}+y\right)\)

Step-by-Step Solution

Answer

\( \ln \left( \frac{ \frac{x}{z} + x }{ \frac{y}{z} + y } \right) \)

Step 1: Understand the given expression

The exercise requires us to express the given logarithmic expression as one single logarithm. The expression is \( \ln \left(\frac{x}{z}+x\right) - \ln \left(\frac{y}{z}+y\right) \).

Step 2: Apply the property of logarithms

Recall the logarithmic property: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). We can apply this property to the given expression.

Step 3: Substitute the property into the expression

Using the property from Step 2, substitute \( a = \frac{x}{z} + x \) and \( b = \frac{y}{z} + y \) into the formula: \[ \ln \left( \frac{x}{z} + x \right) - \ln \left( \frac{y}{z} + y \right) = \ln \left( \frac{\left(\frac{x}{z} + x\right)}{\left(\frac{y}{z} + y\right)} \right) \].

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is essential when working with logarithmic expressions. The most fundamental property is the product rule, which states that the logarithm of a product is the sum of the logarithms of the factors:

• \( \log_b (mn) = \log_b m + \log_b n \).

Another crucial rule is the quotient rule. It tells us that the logarithm of a quotient is the difference of the logarithms:

• \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \).

The power rule allows us to write the logarithm of a power as a product:

• \( \log_b (m^n) = n \cdot \log_b m \).

Knowing these properties makes it easier to manipulate and simplify complex logarithmic expressions in mathematics. They form the foundation of much of what we do with logarithms.

Simplifying Logarithms

Simplifying logarithms is often about applying the right properties effectively. In our exercise, we used the logarithm subtraction property. This property allows us to transform a subtraction of two logarithms into a single logarithm by converting it into the logarithm of the quotient.In simpler terms, if you have two expressions, and you subtract their logarithms, you can express it as a single log by dividing the two expressions inside the logarithm:

• Given: \( \ln(a) - \ln(b) \)
• Simplify as: \( \ln\left(\frac{a}{b}\right) \)

This kind of simplification is useful for condensing long and sometimes confusing logarithmic expressions into more manageable forms. By doing so, you're making the expression easier to evaluate or further manipulate.

Mathematical Expressions

When dealing with expressions in mathematics, especially logarithmic ones, structuring them efficiently is advantageous. A mathematical expression can consist of numbers, variables, and operators. Expressions convey a single numerical value or function when fully evaluated.In logarithmic expressions, combining terms through properties like the product or quotient rule helps in condensing the expression into a single, more straightforward logarithm.For example, transforming \( \ln \left(\frac{x}{z} + x\right) - \ln \left(\frac{y}{z} + y\right) \) into \( \ln \left(\frac{\left(\frac{x}{z} + x\right)}{\left(\frac{y}{z} + y\right)}\right) \) is a way to compact information. It makes the result more straightforward to analyze or compute in subsequent steps. Good mathematical formulation involves knowing when and how to apply such rules, maintaining accuracy, and simplifying where possible to ensure a cleaner, more readable representation.