Problem 84 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(\ln \left(x y+y^{2}\right)-\ln (x z+y z)+\ln z\)

Step-by-Step Solution

Answer

The expression is \( \ln\left(\frac{z(xy + y^2)}{xz + yz}\right) \).

Step 1: Understand the expression

We are given the expression \( \ln(xy + y^2) - \ln(xz + yz) + \ln z \). Our goal is to combine these into a single logarithmic expression.

Step 2: Apply logarithmic rules

Remember that \( \ln(a) - \ln(b) \) can be rewritten as \( \ln\left(\frac{a}{b}\right) \) and \( \ln(a) + \ln(b) \) can be rewritten as \( \ln(ab) \). This means we can start combining the terms.

Step 3: Simplify using subtraction of logs

Use the rule \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \) to simplify the expression to \( \ln\left(\frac{xy + y^2}{xz + yz}\right) + \ln z \).

Step 4: Simplify using addition of logs

Now, apply the rule \( \ln(a) + \ln(b) = \ln(ab) \) to combine the terms into one log: \( \ln\left(z \cdot \frac{xy + y^2}{xz + yz}\right) \).

Step 5: Simplify the final expression

This simplifies the expression to \( \ln\left(\frac{z(xy + y^2)}{xz + yz}\right) \). Thus, the entire expression as one logarithm is \( \ln\left(\frac{z(xy + y^2)}{xz + yz}\right) \).

Key Concepts

Logarithm Properties

Logarithms are mathematical operations that help us solve exponential equations. They have unique properties that make them incredibly useful for simplifying expressions. The most relevant properties here are:

• **Product Rule:** This rule states that the logarithm of a product is the sum of logarithms: \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• **Quotient Rule:** This property indicates that the logarithm of a quotient is the difference of logarithms: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• **Power Rule:** According to this rule, the logarithm of a power can be expressed as the exponent times the logarithm: \( \log_b(m^n) = n \cdot \log_b(m) \).

These properties allow us to rewrite complex logarithmic expressions in simpler or more condensed forms. Understanding these rules is crucial for manipulating and combining logarithms effectively.

Natural Logarithm

The natural logarithm, abbreviated as \( \ln \), is a special kind of logarithm that uses the base \( e \), where \( e \approx 2.718 \). The base \( e \) is an irrational number often found in calculations involving growth rates, such as in finance, physics, and biology.
The natural logarithm has the same properties as other logarithms but is more prevalently used in calculus and advanced mathematics:

• **Equivalent Form:** \( \ln(e^x) = x \) because raising \( e \) to the power of \( x \) and taking the logarithm base \( e \) effectively cancel each other out.
• **Inverse Relationship:** \( \ln(e) = 1 \). Any process involving the natural logarithm and \( e \) can be simplified significantly with this knowledge.

Understanding the behavior of the natural logarithm is vital for simplifying expressions involving \( e \) and for tackling calculus problems.

Expression Simplification

Simplifying expressions is about turning complex problems into easier ones by using mathematical rules. For logarithmic expressions, like the one given, the goal is to use the logarithm properties to reduce them to a single, simpler form.
To simplify the expression \( \ln(xy + y^2) - \ln(xz + yz) + \ln z \), follow these steps:

• **Apply Quotient Rule:** Start by applying the formula \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \) to create \( \ln\left(\frac{xy + y^2}{xz + yz}\right) \).
• **Combine Using Product Rule:** Next, add the logarithms with the product property \( \ln(a) + \ln(b) = \ln(ab) \), resulting in \( \ln\left(z \cdot \frac{xy + y^2}{xz + yz}\right) \).
• **Final Simplification:** Simplify to \( \ln\left(\frac{z(xy + y^2)}{xz + yz}\right) \).

The final result is a streamlined version of the expression, making it easier to interpret or further manipulate if needed.