Problem 20 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt[7]{x} $$

Step-by-Step Solution

Answer

The expansion of \( \ln \sqrt[7]{x} \) is \( \frac{1}{7} \cdot \ln x \)

Step 1: Recognize Logarithmic Property

According to properties of logarithms, \( \ln a^b = b \cdot \ln a \). Therefore, we can apply this property when the logarg (the expression inside the log) is a power, as in our case where the logarg is \( \sqrt[7]{x} \), which is essentially \( x^{1/7} \).

Step 2: Apply Logarithmic Property

By applying the property mentioned before, the expression \( \ln \sqrt[7]{x} \) can be rewritten as \( \frac{1}{7} \cdot \ln x \)

Key Concepts

Expand Logarithmic Expressions

When faced with a complex logarithmic expression, breaking it down into simpler parts can make it easier to work with. This process is called expanding. By using the properties of logarithms, such as the power rule, we can expand expressions which involve roots or powers.

Let's consider an example:

• The expression \(\ln \sqrt[7]{x} \).

First, recognize that \(\sqrt[7]{x} \) can be rewritten as \(x^{1/7} \). This allows us to apply the power rule of logarithms, which states that \(\ln a^b = b \cdot \ln a \). By applying this rule, the expression \(\ln x^{1/7} \) can be expanded to \(\frac{1}{7} \cdot \ln x \).

Expanding expressions like this can simplify complex problems, making them easier to understand and solve systematically. This method is particularly useful in calculus and higher-level mathematics.

Evaluate Logarithmic Expressions

Evaluating logarithmic expressions involves finding their numerical value. While some expressions can be simplified or evaluated exactly, others may require estimation methods or calculators.

• For example, consider the expression \(\ln(x)\), where \(x\) is a number you're familiar with, such as \(x = e^2\).

Here, evaluating \(\ln(e^2)\) is straightforward because it simplifies to \(2 \cdot \ln e\). Knowing that \(\ln e = 1\), the expression further simplifies to 2, giving us the exact evaluation.

This tells us that understanding the fundamental values, like \(\ln 1 = 0\) and \(\ln e = 1\), will facilitate quicker evaluation of expressions without always reaching for a calculator. Knowing properties of numbers associated with logarithms helps in evaluating these expressions efficiently.

Logarithmic Property

The properties of logarithms are powerful tools that help us manipulate and simplify expressions. These include the power rule, as seen before, along with others like the product and quotient rules.

• The product rule states \(\ln(ab) = \ln a + \ln b\).
• The quotient rule states \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
• Transformations using these properties often involve simplification that makes complex expressions manageable.

For example, if we have an expression involving a log of a quotient, like \(\ln\left(\frac{x}{y}\right)\), we can use the quotient rule to break it down into \(\ln x - \ln y\). Similarly, with products, as in \(\ln(2x)\), the product rule simplifies this to \(\ln 2 + \ln x\).

These properties are essential in higher mathematical studies as they simplify the handling of exponential growth and decay problems and integrate into calculus seamlessly.