Problem 18 - 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. $$ \log M^{-8} $$

Step-by-Step Solution

Answer

-8 \cdot \log M

Step 1: Identify the property of logarithms to apply

Observe the given expression \( \log M^{-8} \). Here we have a logarithm of a number raised to a power. We can apply the power rule of logarithms.

Step 2: Apply the power rule

According to the power rule \( \log_b M^n = n \cdot \log_b M \). Thus, we can express \( \log M^{-8}\) as \( -8 \cdot \log M \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are crucial components in mathematics, representing the power to which a base number must be raised to obtain a certain number. When we deal with these expressions, we're often manipulating them to simplify or to express them in a different form.
One of the key purposes of performing operations on logarithmic expressions is to make problems more manageable by converting complex multiplications or divisions into additions or subtractions.
For example, these expressions are widely used in exponential growth models, such as population growth and radioactive decay.

• Tackles complex numbers in exponential form
• Useful in solving exponential equations

Understanding logarithms is foundational in mathematics as it extends to various fields like computer science, physics, and engineering. Logarithmic expressions such as \( \), which are familiar forms, use the notation \( \log_b M \), where \( b \) is the base, and \( M \) is the number you're taking the logarithm of.

Power Rule of Logarithms

The power rule of logarithms is one of the fundamental properties that simplify the manipulation of logarithmic expressions. This rule states that for any logarithm \( \log_b M^n \), you can "bring down" the exponent \( n \) and multiply it to the log expression, resulting in \( n \cdot \log_b M \).
This is particularly useful when dealing with powers, making it easier to handle complicated expressions by transforming them into simpler terms.

• Simplifies logarithms of powers
• Transforms multiplication into addition, allowing easier computations

Consider the expression \( \log M^{-8} \). By applying the power rule, the expression can be expanded to \( -8 \cdot \log M \). This property can be used to break down more intricate calculations by leveraging similar transformations. Remember, the power rule is applicable for any real number exponent, positive or negative.

Expand Logarithmic Expressions

Expanding logarithmic expressions involves using various properties of logarithms to break down a complex logarithm into simpler, more manageable parts. This step often utilizes several logarithmic rules, including the product rule, quotient rule, and especially the power rule.
For instance, the expression \( \log M^{-8} \) can be expanded using the power rule to \( -8 \cdot \log M \). This transformation simplifies the expression, converting exponential components into straightforward, multiplicative terms.

• Breaks down complex expressions
• Utilizes properties for simplification

Such processes are vital in solving logarithmic equations more effectively, understanding data transformations, and optimizing mathematical models. The power of expansion lies in how it makes complex multiplications or divisions in logarithmic form much simpler and therefore easier to analyze or compute, especially when seeking further information or when preparing to solve equations.