Problem 36 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \sqrt{\frac{x^{3}}{x+1}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3}{2} \ln{x} - \frac{1}{2} \ln{(x+1)}\).

Step 1: Expressing the square root as a power

The square root operator can be expressed as the power of \(\frac{1}{2}\), resulting in \(\ln (\frac{x^{3}}{x+1})^{\frac{1}{2}}\).

Step 2: Apply the power rule of logarithms

The power rule states that \(\log_b{m^n} = n \log_b{m}\). Applying this rule to our expression we get \(\frac{1}{2} \ln (\frac{x^{3}}{x+1})\).

Step 3: Apply the quotient rule of logarithms

The quotient rule states that \(\log_b{\frac{m}{n}} = \log_b{m} - \log_b{n}\). Applying this rule to our expression we get \(\frac{1}{2} (\ln{x^{3}} - \ln{x+1})\).

Step 4: Apply the power rule again

Applying the power rule again to \(\ln{x^{3}}\) in the previous expression, we finally get \(\frac{3}{2} \ln{x} - \frac{1}{2} \ln{(x+1)}\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms can greatly simplify complex expressions. Applying these properties systematically can turn intricate operations into simpler tasks. Here are some key properties that are often used:

• Product Rule: The product rule states that the logarithm of a product is the sum of the logarithms: \(\log_b{(mn)} = \log_b{m} + \log_b{n}\).
• Quotient Rule: The quotient rule states that the logarithm of a division is the difference of the logs: \(\log_b{\frac{m}{n}} = \log_b{m} - \log_b{n}\).
• Power Rule: According to the power rule, the log of a power is the exponent times the log of the base: \(\log_b{m^n} = n \log_b{m}\).

These rules are essential when working with logarithmic expressions. Each property allows for the expression to be rewritten in a different, often simpler form. In our original expression, we use the quotient rule and power rule multiple times to achieve the desired simplification.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms to transform and simplify various mathematical problems. At first glance, these expressions may seem daunting. However, they become manageable once broken down using the properties mentioned before. Let's explore how our original expression \(\ln \sqrt{\frac{x^{3}}{x+1}}\) can be treated effectively:

• First, transform the square root into a fractional exponent, which is often easier to handle with logarithms. Thus, \(\sqrt{\frac{x^{3}}{x+1}}\) becomes \((\frac{x^3}{x+1})^{\frac{1}{2}}\).
• Utilize the power rule by bringing down the exponent: \(\frac{1}{2} \ln (\frac{x^3}{x+1})\).
• Apply the quotient rule to separate the expression into \(\ln x^3\) and \(\ln (x+1)\) resulting in \(\frac{1}{2} (\ln{x^3} - \ln{(x+1)})\).

By manipulating the logarithmic expressions using these principles, a seemingly complex equation becomes comprehensible and easier to work with.

Algebraic Manipulation

Algebraic manipulation is a powerful tool in simplifying expressions, especially when dealing with logarithms. The ability to rework an expression into a more convenient form is crucial for solving logarithmic equations. Here’s how we break down the original exercise through algebraic manipulation:

• Begin by expressing the square root as a power of \(\frac{1}{2}\), allowing you to use the power rule of logarithms efficiently.
• Next, use the power rule to isolate the exponent outside the logarithm, simplifying calculations.
• Finally, apply the quotient rule to split the log expression into subtractive parts, making further simplifications possible.

For instance, in the step-by-step solution, after using the power rule, the expression \(\frac{1}{2} \ln (\frac{x^3}{x+1})\) is formed. This acts as an intermediate step, simplifying the problem before applying the quotient rule. Each manipulation step transforms the expression towards its simplest form, which reveals the true potential and practicality of algebraic manipulation in simplifying logarithmic equations.