Problem 8 - College Algebra

Question

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. $$ \log _{2}\left(y^{x}\right) $$

Step-by-Step Solution

Answer

\( x \cdot \log_2(y) \)

Step 1: Identify the Expression

We expand the logarithmic expression:
For the following exercises, expand each logarithm as much as possible.
Rewrite each expression as a sum, difference, or product of logs.
$$
\log _{2}\left(y^{x}\right)
$$

Step 2: Apply Logarithm Rules

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b(M/N) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^p) = p\log_b(M)\)

Step 3: Expanded Expression

\( x \cdot \log_2(y) \)

Key Concepts

Power Rule of Logarithms

When working with logarithmic expressions, one crucial property to understand is the Power Rule of Logarithms. This rule simplifies expressions where the logarithm is taken of a number raised to a power. The Power Rule states that \( \log_b(a^c) = c \cdot \log_b(a) \). This means if you have a logarithm of a power, you can "bring down" the exponent as a multiplier in front of the logarithm.

For example, consider the expression \( \log_2(y^x) \). According to the Power Rule, you can move the exponent \( x \) in front of the logarithm, transforming the expression into \( x \cdot \log_2(y) \).


This rule is particularly useful because it allows us to simplify logarithmic expressions and make them easier to work with, especially when they involve complex or large powers. Understanding and applying the Power Rule is fundamental to mastering more advanced logarithmic operations.

Logarithm Properties

Logarithms have a set of properties that are essential for simplifying and manipulating expressions. These properties are akin to the rules we often see in traditional algebra, but they operate on this specific type of mathematical expression.

Here are the key properties of logarithms:

• Product Rule: This rule states that \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \). It signifies that the logarithm of a product can be split into the sum of the logarithms.
• Quotient Rule: This rule is expressed as \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \), showing that the logarithm of a quotient is the difference of the logarithms.
• Change of Base Formula: Often used for simplifying calculations or switching bases, this is \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \).

These properties are incredibly helpful for rewriting logarithmic expressions in simpler forms. Applying these can often lead to solutions that are easier to understand and compute.

Expanding Logarithms

Expanding logarithms refers to the process of breaking down a single logarithmic expression into multiple terms that may involve sums, differences, or products of logs. This is achieved using the properties of logarithms, such as the Power Rule and others mentioned.

To expand a logarithmic expression like \( \log_2(y^x) \), we initially apply the Power Rule, transforming it to \( x \cdot \log_2(y) \). If the term \( y \) is itself a product or quotient of numbers or expressions, further expansion can occur using the Product and Quotient Rules. For instance, if \( y \) were \( m \cdot n \), this expression could be further expanded to \( x \cdot (\log_2(m) + \log_2(n)) \).

Expanding logarithms often makes it easier to differentiate and integrate functions involving logarithms, solve equations, and perform other algebraic manipulations. It turns complex logarithmic expressions into a collection of simpler ones, making the calculations more manageable.