Problem 30 - College Algebra with Modeling and Visualization

Question

Use a calculator to approximate each pair of expressions. Then state which property of logarithms this calculation illustrates. $$ \log \frac{\sqrt{x^{2}+4}}{\sqrt[3]{x-1}} $$

Step-by-Step Solution

Answer

The calculation illustrates the subtraction property of logarithms.

Step 1: Break Down the Expression

First, let's break down the given expression: \( \log \frac{\sqrt{x^{2}+4}}{\sqrt[3]{x-1}} \). This can be rewritten using logarithmic properties as \( \log \left( \sqrt{x^2 + 4} \right) - \log \left( \sqrt[3]{x - 1} \right) \).

Step 2: Apply the Power Rule

We apply the power rule of logarithms, which is \( \log \left(a^b\right) = b \log a \). For the first term, \( \log \left(\sqrt{x^2 + 4} \right) = \log \left((x^2 + 4)^{1/2}\right) = \frac{1}{2} \log (x^2 + 4) \). For the second term, \( \log \left(\sqrt[3]{x - 1}\right) = \log \left((x - 1)^{1/3}\right) = \frac{1}{3} \log (x - 1) \).

Step 3: Rewrite Using Logarithmic Subtraction Property

Now rewrite the expression as follows: \( \frac{1}{2} \log (x^2 + 4) - \frac{1}{3} \log (x - 1) \). This expression illustrates the subtraction property of logarithms, which states that \( \log \frac{a}{b} = \log a - \log b \).

Step 4: Use Calculator for Numerical Approximation

Assuming a value for \( x \) such that \( x > 1 \), we calculate each part separately using a calculator. For instance, if \( x = 2 \), \( \log \frac{\sqrt{4+4}}{\sqrt[3]{2-1}} = \log \left(\sqrt{8}\right) - \log \left( \sqrt[3]{1} \right) \), which equals \( \log 2.828 - 0 = 0.4515 \) (approximate values via calculator).

Key Concepts

Properties of Logarithms

Logarithms might seem tricky at first, but they follow a set of basic properties that can make them easier to understand and work with. One of the most useful properties is the subtraction property. This property states that the logarithm of a division can be rewritten as the subtraction of two separate logarithms:

\[ \log \frac{a}{b} = \log a - \log b \]

This is exactly what we used in the step-by-step solution. We transformed the original expression, \( \log \frac{\sqrt{x^{2}+4}}{\sqrt[3]{x-1}} \), into something more manageable:

\[ \log \left( \sqrt{x^2 + 4} \right) - \log \left( \sqrt[3]{x - 1} \right) \]

Other key logarithmic properties include:

• Product Property: \( \log(ab) = \log a + \log b \)
• Power Rule: \( \log(a^b) = b \log a \), which we used to simplify the expression involving roots.

These properties are very handy and can simplify complex logarithmic expressions into simpler ones that are easier to evaluate or approximate.

Logarithmic Identities

Understanding logarithmic identities can greatly enhance your mathematical flexibility. These identities often build the foundation for more complex operations. The Power Rule of logarithms is one such vital identity. It allows us to take an exponent out of the logarithm:

\[ \log(a^b) = b \log a \]

In our given exercise, this identity helped simplify the expression:\[ \log \left( \sqrt{x^2 + 4} \right) = \frac{1}{2} \log (x^2 + 4) \]and\[ \log \left( \sqrt[3]{x - 1} \right) = \frac{1}{3} \log (x - 1) \]

These transformations make the expressions more manageable while keeping them mathematically equivalent. Remember also the change of base formula, which comes in handy for calculations:

\[ \log_b a = \frac{\log_c a}{\log_c b} \]

Each identity offers a pathway to manipulate log forms, leading to easier calculations or deeper insights into a problem's structure.

Mathematical Approximation

When dealing with logarithms, sometimes exact values aren't possible due to irrational numbers or complex calculations. This is where approximation comes into play. Calculators can provide numerical values for logarithmic expressions that would otherwise be cumbersome to establish.

For instance, in the exercise, choosing \(x = 2\) made the expression involve easier numbers, like \(\log \sqrt{8} - \log 1\). Using this method, we reached an approximate value of 0.4515.

Here’s how to approach approximation:

• Convert complex expressions into simpler forms using identities and properties.
• Select a value for variables that makes sense in context (often simpler values work best).
• Use a calculator to find decimal approximations to gain intuitive understanding.

Remember, mathematical approximation is a practical skill that helps especially when dealing with real-world applications and when exact calculations are not feasible.