Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to expand and simplify the expression. $$\log x\left(x^{2}+1\right)^{-1 / 2}$$

Step-by-Step Solution

Answer

The expanded and simplified expression for \(\log x\left(x^{2}+1\right)^{-\frac{1}{2}}\) is \(\log x + \frac{1}{2} \log\left(x^{2}+1\right)\).

Step 1: Apply the Power Rule

We have the expression \(\log x(x^{2}+1)^{-\frac{1}{2}}\). First, let's apply the Power Rule to the second factor, \((x^{2}+1)^{-\frac{1}{2}}\), to rewrite the expression: \(\log x\left[\left(x^{2}+1\right)^{-\frac{1}{2}}\right] = \log x[-\frac{1}{2}\log\left(x^{2}+1\right)]\) Now our expression looks like \(\log x[-\frac{1}{2}\log\left(x^{2}+1\right)]\)

Step 2: Apply the Product Rule

Next, let's apply the Product Rule to the two logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. \(\log x[-\frac{1}{2}\log\left(x^{2}+1\right)] = \log x - \left(-\frac{1}{node2} \log\left(x^{2}+1\right)\right)\) Now our expression looks like \(\log x + \frac{1}{2} \log\left(x^{2}+1\right)\)

Step 3: Simplification

The expression is now simplified and expanded as per the logarithmic properties discussed earlier. \(\log x\left(x^{2}+1\right)^{-\frac{1}{2}}\) = \(\log x + \frac{1}{2} \log\left(x^{2}+1\right)\) So, the expanded and simplified expression is \(\log x + \frac{1}{2} \log\left(x^{2}+1\right)\).

Key Concepts

Power Rule in Logarithms

The Power Rule is a fundamental logarithmic concept that helps us simplify expressions where variables are raised to powers within logarithms.

• This rule states that \[ ext{If } y = b^x, ext{ then } \ ext{log}_b(y^n) = n imes ext{log}_b(y). \]
• Simply put, exponents in the arguments of a logarithm can be pulled out front as a multiplier.

In the context of the problem, we started with \[ ext{log} ig( x(x^2 + 1)^{-1/2} ig). \] Here, we can use the Power Rule for the term \( (x^2 + 1)^{-1/2} \). The Power Rule allows us to rewrite \( -\frac{1}{2} \log(x^2 + 1) \), emphasizing how exponents simplify our expression by translating powers into multipliers in front of the logarithm. Using this powerful tool, we initiated a transformation that gave us a simpler expression derived from the original exponent.

Understanding the Product Rule

The Product Rule is another key feature in handling logarithmic expressions, particularly when dealing with products within a logarithm.

• This rule states that\[ ext{log}_b(m imes n) = ext{log}_b(m) + ext{log}_b(n). \]
• Essentially, the logarithm of a product is the sum of the logarithms of its individual factors.

For the given problem, after applying the Power Rule, our next task was to deal with the expression\( \log x[-\frac{1}{2}\log(x^2 + 1)] \) using the Product Rule. This allows us to rewrite it as \( \log x + \frac{1}{2} \log(x^2 + 1). \) The Product Rule effectively breaks down more complicated product expressions into simpler additive logarithmic expressions. By breaking down the original expression into these components, we streamline the process of simplification and achieve the expanded expression.

Logarithmic Properties at Work

To effectively expand and simplify expressions involving logarithms, understanding various logarithmic properties is crucial. These properties help break down complex expressions systematically into manageable forms.

• We have already discussed the Power Rule and the Product Rule, which are integral to our problem.
• Other useful properties include the Quotient Rule, \[ \text{log}_b\left(\frac{m}{n}\right) = \text{log}_b(m) - \text{log}_b(n), \]which helps handle divisions in logarithms, though not directly used here.
• The Change of Base Formula might also come handy when converting logarithms to different bases, stated as\[ \text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)}, \]useful for simplifying calculations in multi-base logarithms.

All these properties share a common goal: simplifying complex expressions by breaking them down into basic logarithmic terms. Knowing these rules enhances our ability to manipulate and expand logarithmic expressions efficiently, leading to solutions that are clear and precise. In our example, a strategic application of these properties yielded the neatly expanded expression \( \text{log} x + \frac{1}{2} \text{log}(x^2 + 1) \). Grasping these properties provides a toolkit for tackling a wide array of logarithmic problems smoothly.