Problem 80 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \left(\frac{x}{\sqrt{x^{2}+1}}\right)\)

Step-by-Step Solution

Answer

The expanded form of \( \ln \left(\frac{x}{\sqrt{x^{2}+1}}\right) \) is \( \ln (x) - \frac{1}{2} \ln (x^{2}+1) \)

Step 1: Apply the quotient rule

The quotient rule states that the logarithm of a quotient is the difference of the logarithms. So, we can write \( \ln \left(\frac{x}{\sqrt{x^{2}+1}}\right) \) as \( \ln (x) - \ln (\sqrt{x^{2}+1}) \)

Step 2: Apply the power rule

The power rule allows us to pull down the exponent in a logarithm as a multiplier. We use this to simplify \( \ln (\sqrt{x^{2}+1}) \) to \( \frac{1}{2} \ln (x^{2}+1) \)

Step 3: Simplify the expression

Since we've successfully broken down the original expression to simpler logarithmic expressions, our final expanded expression becomes \( \ln (x) - \frac{1}{2} \ln (x^{2}+1) \)

Key Concepts

Properties of Logarithms

Logarithms are mathematical functions that help us work with large numbers by turning multiplication into addition and division into subtraction. Understanding the properties of logarithms is crucial for simplifying and expanding logarithmic expressions.
Some key properties include:

• Product Rule: The logarithm of a product is the sum of the logarithms. If you have \(\log_b(M\cdot N)\), it is equal to \(\log_b(M) + \log_b(N)\).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms. For \(\log_b\left(\frac{M}{N}\right)\), it simplifies to \(\log_b(M) - \log_b(N)\).
• Power Rule: The logarithm of a power allows the exponent to be multiplied out front. Thus, \(\log_b(M^k)\) becomes \(k \cdot \log_b(M)\).
• Logarithm of 1: Any logarithm of 1 with a positive base is always 0 - \(\log_b(1) = 0\).

These properties are foundational in various mathematical fields and are pivotal for transforming complex logarithms into simpler forms.

Quotient Rule

The Quotient Rule is a powerful tool for simplifying logarithmic expressions that involve division. It transforms the logarithm of a quotient into the difference of two separate logarithms.
For example, the expression \(\ln\left(\frac{x}{\sqrt{x^2+1}}\right)\) falls into this category. By applying the Quotient Rule, we rewrite it as \(\ln(x) - \ln(\sqrt{x^2+1})\).
Here's a simple way to understand it:

• The numerator of the fraction becomes the first part of your difference.
• The denominator becomes the second part of your difference.

This method is particularly useful when you need to break down complex division operations in logarithmic terms, turning them into more manageable sub-problems.

Power Rule

The Power Rule in logarithms allows us to simplify expressions where the argument of the logarithm is raised to a power. This is done by taking the exponent and moving it out front as a coefficient, thereby reducing the complexity of the original expression.
For example, in the expression \(\ln(\sqrt{x^2+1})\), the square root can be written as a power: \(\ln((x^2+1)^{1/2})\). Applying the Power Rule, we move the \(1/2\) outside the logarithm, resulting in \(\frac{1}{2}\ln(x^2+1)\).
This transformation makes it easier to work with the logarithmic expressions, especially when calculating, integrating, or deriving them.

• The power in the argument becomes a multiplier upfront.
• This process simplifies complex expressions with exponents.

The Power Rule is integral for expanding and simplifying logarithmic expressions that involve powers.

Logarithmic Expression Simplification

Simplifying logarithmic expressions involves using various properties of logarithms to transform a complex expression into a more manageable form. This is especially important in calculus, algebra, and other math disciplines where simplification is often necessary to proceed with solving equations.
To simplify the expression like \(\ln\left(\frac{x}{\sqrt{x^2+1}}\right)\), we follow a methodological approach:

• Step 1: Identify components of the expression that can be simplified using known properties. Begin by applying the Quotient Rule to handle division.
• Step 2: If applicable, use the Power Rule to simplify parts of the expression that contain exponents, as seen with the square root.
• Step 3: Combine these simplified parts into the final expression which is easier to interpret or solve.

After simplification, we obtain an expression like \(\ln(x) - \frac{1}{2}\ln(x^2+1)\), which is more straightforward and easier to handle for further manipulation.