Problem 19 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expanded form is \( 1 + \log_{2}(x) \).

Step 1: Identify the Expression

We are given the expression \( \log_{2}(2x) \). Our goal is to expand this expression using the laws of logarithms.

Step 2: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \log_b(mn) = \log_b(m) + \log_b(n) \). Apply this rule to \( \log_{2}(2x) \). We can rewrite it as: \[ \log_{2}(2x) = \log_{2}(2) + \log_{2}(x) \].

Step 3: Simplify \(\log_{2}(2)\)

To further simplify, evaluate \( \log_{2}(2) \). Since any base to the power of 1 is the base itself, \( \log_{2}(2) = 1 \).

Step 4: Write the Final Expanded Form

Substitute the simplified value of \( \log_{2}(2) \) back into the expression: \[ \log_{2}(2x) = 1 + \log_{2}(x) \].

Key Concepts

Product Rule of Logarithms

The Product Rule of Logarithms is a crucial concept useful for simplifying complex logarithmic expressions. It essentially transforms a log of a product into the sum of logs. This rule states that for any positive numbers \(m\) and \(n\), and any positive base \(b\) (where \(b eq 1\)), the logarithm of the product of \(m\) and \(n\) is equal to the sum of the logarithms of \(m\) and \(n\) with the same base:

• \( \log_b(mn) = \log_b(m) + \log_b(n) \)

To understand this, consider the expression \( \log_2(2x) \). Here, the terms "2" and "x" are multiplied inside the logarithm. By applying the product rule, we can split the log into two parts:

\[ \log_2(2x) = \log_2(2) + \log_2(x) \].

This transformation not only makes the expression easier to handle, but also brings clarity when working with more complicated logarithmic equations. Remember, simplifying logs using this rule is always applied to expressions where multiplication is inside the logarithm.

Logarithmic Expansion

Logarithmic Expansion involves breaking down a complex logarithmic expression into simpler terms using logarithmic properties like the product, quotient, and power rules. Expanding expressions makes them easier to analyze and manipulate.

Take the expression \( \log_2(2x) \) as an example. The goal of expansion here is to express this in simpler terms. By applying the Product Rule of Logarithms, we expand the expression into multiple terms:

• \( \log_2(2x) = \log_2(2) + \log_2(x) \)

Next, we further simplify the expression by evaluating parts of it. Since \( \log_2(2) \) simplifies to 1 (because 2 raised to the power of 1 is 2), the expression becomes:

\[ \log_2(2x) = 1 + \log_2(x) \].

By expanding the expression, we achieve clarity and can better understand its components, making it easier to use in diverse mathematical situations.

Logarithmic Simplification

Logarithmic Simplification is the process of reducing a logarithmic expression to its simplest form. This process can involve evaluating constants, combining terms, and applying different logarithmic rules.

Consider the expression \( \log_2(2x) \). After applying the Product Rule and expanding the expression to \( \log_2(2) + \log_2(x) \), the next step is to simplify. Notably, \( \log_2(2) \) equals 1, because 2 is the base, and 2 raised to the power of 1 is equal to itself:

• \( \log_2(2) = 1 \)

Hence, the expanded expression simplifies from:

\[ \log_2(2x) = \log_2(2) + \log_2(x) \]

to:

• \( 1 + \log_2(x) \)

This final form is much simpler and more user-friendly. Simplification helps ensure that working with logarithms is straightforward and gives us more accurate and clear results in calculations and applications.