Problem 22 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expanded expression is: \( 10 \cdot \log_{2}(x) + 10 \cdot \log_{2}(y) \).

Step 1: Apply the Power Rule

The Power Rule of Logarithms states that \( \log_b(M^n) = n \cdot \log_b(M) \). In this expression, we can take the 10 out as a coefficient: \[ \log_{2}((xy)^{10}) = 10 \cdot \log_{2}(xy) \].

Step 2: Apply the Product Rule

The Product Rule of Logarithms states that \( \log_b(MN) = \log_b(M) + \log_b(N) \). Apply this rule to \( \log_{2}(xy) \): \[ \log_{2}(xy) = \log_{2}(x) + \log_{2}(y) \].

Step 3: Expand using the coefficients

Substitute the expanded form from Step 2 back into Step 1's equation: \[ 10 \cdot (\log_{2}(x) + \log_{2}(y)) = 10 \cdot \log_{2}(x) + 10 \cdot \log_{2}(y) \].

Key Concepts

Power Rule

The Power Rule is a fundamental concept in the realm of logarithms, often used to simplify expressions with exponents. According to this rule, if you have a logarithm of a number raised to an exponent, you can "bring down" that exponent as a multiplier in front of the logarithm. Mathematically, the Power Rule is expressed as:
\[\log_b(M^n) = n \cdot \log_b(M)\]This can be really handy when you're trying to expand or simplify logarithmic expressions.

• Imagine you have \( \log_2((xy)^{10}) \): here, the entire expression \((xy)\) is raised to the 10th power.
• To simplify it, apply the Power Rule: you "take out" the 10 to multiply the logarithm, which makes it \( 10 \cdot \log_2(xy) \).

This first step not only makes the expression less cumbersome but sets the stage for further expansion.

Product Rule

The Product Rule is another useful tool when dealing with logarithmic expressions. It lets you handle products inside a logarithm by turning them into a sum of two separate logarithms. Here's how it goes:
\[\log_b(MN) = \log_b(M) + \log_b(N)\]This transformation is particularly helpful when expanding expressions involving products. For example:

• Once you've applied the Power Rule to our example and have \( 10 \cdot \log_2(xy) \), the Product Rule is your next stop.
• Apply it to split \( \log_2(xy) \) into \( \log_2(x) + \log_2(y) \).

Now, instead of one complex term, you have a straightforward sum, which you can easily manage in calculations or further manipulations.

Logarithmic Expansion

The final step in almost any problem involving the expansion of logarithmic expressions is to rewrite the expression in its fully expanded form. After applying the Power and Product Rules, you're usually left with a mix of separated terms. In our example:

• We ended with \( 10 \cdot (\log_2(x) + \log_2(y)) \).
• By distributing the 10, we get \( 10 \cdot \log_2(x) + 10 \cdot \log_2(y) \).

This process of expanding allows you to break down complicated logarithmic statements into more simplistic, manageable parts. It's like unpacking a suitcase and organizing everything into tidy stacks. Besides making complex expressions more tractable, it can aid in solving equations, integrating, or differentiating logarithmic functions. Each small sum of separate logarithms represents part of the original puzzle.