Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt{z}$$.

Step-by-Step Solution

Answer

\( \frac{1}{2} \ln z \)

Step 1: Identify the logarithm property

Identify the property that the logarithm of a root is equal to the logarithm divided by the root's degree. This means that \( \ln \sqrt{z} \) can be expressed as \( \frac{1}{2} \ln z \).

Step 2: Apply the logarithm property

By applying the identified property, expand \( \ln \sqrt{z} \) as \( \frac{1}{2} \ln z \). This is the required expression in terms of sum, difference or constant multiple of logarithms.

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms, which help us to handle numbers that grow rapidly or to shorten multiplications and divisions into simple additions and subtractions. Logarithms can be perceived as the inverse of exponentiation, meaning they help us answer questions like: "What exponent do we need to raise a certain base to get a specific number?"

Here's a simple breakdown: if we have \( b^x = a \), then the logarithm base \( b \) of \( a \) (written as \( \log_b a \)) will give us \( x \). However, natural logarithms use the base \( e \), where \( e \approx 2.718 \), a fundamental constant in mathematics. Hence, \( \ln a = \log_e a \).

You can expand and simplify logarithmic expressions using various properties. Knowing how to do this is useful for solving complex problems more efficiently and understanding the relationships between different quantities.

Expanding Logarithms

Expanding logarithms using properties turns complex expressions into simpler ones, ideally involving sums, differences, or multiples of logs. This can make equations easier to handle, especially when dealing with complex mathematical problems or data analysis.

Here are some important properties to expand logarithms:

• Product Rule: \( \log_b(mn) = \log_b m + \log_b n \)
• Quotient Rule: \( \log_b\left( \frac{m}{n} \right) = \log_b m - \log_b n \)
• Power Rule: \( \log_b(m^n) = n \cdot \log_b m \)

Using these properties, complex logarithmic expressions can be simplified into more approachable forms. For instance, the step-by-step solution showcases expanding \( \ln \sqrt{z} \) into a multiple of a logarithm using the power rule. That is, \( \ln \sqrt{z} \) becomes \( \frac{1}{2} \ln z \), where \( \sqrt{z} = z^{1/2} \).

Expanding logs helps in better understanding and solving a wide array of mathematical and practical real-world problems.

Logarithm of a Root

The logarithm of a root is a specific case of the power rule for logarithms. When dealing with roots, we are essentially involved with fractional exponents. This involves a direct application of the power rule for simplifying logarithmic expressions.

For instance, the square root of a variable \( z \), \( \sqrt{z} \), is expressed as \( z^{1/2} \). According to the power rule property \( \log_b(m^n) = n \cdot \log_b m \), the logarithm of \( \sqrt{z} \) becomes \( \frac{1}{2} \ln z \). This shows how the index of the root (in this case 2, as in \( \sqrt{z} \)) turns into a fraction that multiplies the logarithm.

Understanding the logarithm of a root can simplify calculations and expressions when dealing with square, cube, or higher-order roots. This method is invaluable in calculus and advanced algebra, helping to streamline otherwise complicated mathematical expressions. Always ensure to handle roots carefully in logarithmic expressions to achieve an accurate and simplified result.