Problem 8 - Algebra 2

Question

Write each expression as a single natural logarithm. \(\frac{1}{3}(\ln x+\ln y)-4 \ln z\)

Step-by-Step Solution

Answer

\(\ln ((xy)^\frac{1}{3} / z^4)\)

Step 1: Apply the multiplication property to the first part of expression.

Begin by applying the multiplication property of logarithms, which states that \(\ln a + \ln b = \ln (ab)\). By doing this to \(\frac{1}{3}(\ln x+\ln y)\), the equation simplifies to: \(\frac{1}{3}\ln (xy)\). This condenses the two logarithms into one.

Step 2: Simplify the second part of the expression.

The property \(a\ln b = \ln b^a\) can be applied to simplify \(4\ln z\) into \(\ln z^4\). This transforms it into a single logarithm.

Step 3: Final simplify by applying subtraction property.

Finally, we can apply the subtraction property of logarithms which states that \(\ln a - \ln b = \ln \(\frac{a}{b}\) to the expression. The final expression is \(\frac{1}{3}\ln(xy) - \ln (z^4)\), which simplifies down to: \(\ln (\(xy)^\frac{1}{3} / z^4)\)

Key Concepts

Properties of Logarithms

Logarithms have a set of useful properties that make it easier to manipulate and simplify expressions. These properties are based on fundamental rules of exponents, as logarithms are the inverse of exponential functions. Here are some important properties of logarithms you should know:

• Multiplication Property: This property states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, \[ \ln (ab) = \ln a + \ln b \]This is helpful when you want to combine multiple logarithmic terms into a single term.
• Power Property: This property allows you to bring the exponent of an argument to the front of the logarithm. Expressed as:\[ a \ln b = \ln b^a \]It simplifies terms involving powers to make complex logarithms more manageable.
• Division (Subtraction) Property: When working with the division of numbers, the logarithms of the dividend and divisor can be expressed as a subtraction. This property is written as:\[ \ln \left(\frac{a}{b}\right) = \ln a - \ln b \]This property is particularly useful for simplifying expressions with division or subtraction operations involved.

These properties allow for the efficient transformation and simplification of logarithmic expressions, which is critical in solving mathematical problems involving logarithms.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using properties of logarithms to condense multiple logarithmic terms into a single expression. This process often makes it easier to solve equations or perform calculus operations. Here's a practical breakdown of simplifying the given exercise:Given expression: \[ \frac{1}{3}(\ln x+\ln y)-4 \ln z \]Step 1: Use the multiplication property. Combine \( \ln x \) and \( \ln y \) into one term:\[ \frac{1}{3}\ln (xy) \]Step 2: Apply the power property on \( 4 \ln z \), transforming it to:\[ \ln z^4 \]Step 3: Use the division property to express the whole equation as a single logarithm:\[ \ln \left(\frac{(xy)^{\frac{1}{3}}}{z^4}\right) \]The ability to simplify expressions using these steps not only reduces complexity but also helps in gaining insights into the behavior of logarithmic functions.

Logarithmic Functions

Logarithmic functions are a fundamental type of mathematical function used across various fields including science, engineering, and finance. A logarithmic function is typically defined as\[ y = \log_b(x) \]or when using natural logarithms, as\[ y = \ln(x) \]Here, \( b \) is the base of the logarithm, with \( e \) being the base when referring to natural logarithms.Some characteristics to remember about logarithmic functions include:

• Inverse Relationship: Logarithmic functions are the inverse of exponential functions. This means they can be used to "undo" exponentiation. For example, if \( a = b^c \), then \( c = \log_b(a) \).
• Asymptotic Behavior: Graphically, logarithmic functions increase slowly, and they have a vertical asymptote along the y-axis, meaning they approach but never reach a vertical line at \( x = 0 \).
• Usage and Application: They are particularly useful in dealing with multiplication and division operations, turning them into addition and subtraction, which simplifies calculations significantly.

Learning to work with logarithmic functions efficiently can give you powerful tools for mathematical analysis, especially when dealing with exponential growth or decay scenarios. Understanding these concepts deeply enhances your ability to tackle complex mathematical and real-world problems efficiently.