Problem 28 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}(X y)^{10} $$

Step-by-Step Solution

Answer

Expand to \( 10\log_2(X) + 10\log_2(y) \).

Step 1: Apply the Power Rule of Logarithms

The Power Rule states that \( \log_b(M^n) = n \cdot \log_b(M) \). In our expression, \( M = Xy \) and \( n = 10 \). Thus, we have: \[ \log_2((Xy)^{10}) = 10 \cdot \log_2(Xy) \].

Step 2: Apply the Product Rule of Logarithms

The Product Rule states that \( \log_b(MN) = \log_b(M) + \log_b(N) \). Applying this rule to \( \log_2(Xy) \), we get: \[ \log_2(Xy) = \log_2(X) + \log_2(y) \].

Step 3: Substitute Back and Simplify

From Step 1, we have \( 10 \cdot \log_2(Xy) \). Substituting the expanded form from Step 2, this becomes: \[ 10 \cdot (\log_2(X) + \log_2(y)) \]. Distribute the \( 10 \): \[ 10 \cdot \log_2(X) + 10 \cdot \log_2(y) \].

Key Concepts

Power Rule of Logarithms

The Power Rule of Logarithms is an incredibly handy tool for simplifying logarithmic expressions that involve powers. You can think of it as a bridge between exponents and logarithms. The rule states: if you have a logarithm of an exponential term, such as \( \log_b(M^n) \), it can be rewritten by bringing the power in front as a multiplier: \( n \cdot \log_b(M) \).
This property is immensely useful for solving equations where the exponent is making things complicated. By moving the exponent to the front, the equation becomes linear relative to the exponent, making it much easier to handle.

In the exercise above, with \( \log_2((Xy)^{10}) \), the term \((Xy)^{10}\) involves a power of 10. Applying the Power Rule allows us to express it as \( 10 \cdot \log_2(Xy) \), instantly simplifying the expression.

Product Rule of Logarithms

The Product Rule of Logarithms is another powerful property that helps break down complex logarithmic expressions. When two numbers are multiplied inside a logarithm, the rule lets you split them into separate logarithms that are added together. It is stated as: \( \log_b(MN) = \log_b(M) + \log_b(N) \).
This transformation is key for expansion, making it easier to manipulate and simplify expressions that involve multiple factors.

In the given exercise, once we simplified the power using the Power Rule, we needed to work with \( \log_2(Xy) \). Applying the Product Rule, the expression \( \log_2(Xy) \) breaks into \( \log_2(X) + \log_2(y) \), thereby allowing each variable to be treated independently.

Expanding Logarithmic Expressions

Expanding logarithmic expressions is about utilizing properties like the Power Rule and Product Rule to rewrite them in a simplified form. This process breaks down complex logarithms into a series of easier-to-handle terms.
The goal here is clear: Make the expression as straightforward as possible, which in turn facilitates further calculations or even integration into larger formulae.

Let's revisit our exercise: We started with \( \log_2((Xy)^{10}) \). By applying the Power Rule, we recast it as \( 10 \cdot \log_2(Xy) \). Then, using the Product Rule, we expanded it further to \( 10 \cdot (\log_2(X) + \log_2(y)) \). This final step brought it all together as \( 10 \cdot \log_2(X) + 10 \cdot \log_2(y) \).
Such expansions are not just for academic exercises—they are practical techniques used in data science, engineering, and many fields where logarithms abound.