Problem 30 - Algebra 2 and Trigonometry

Question

Expand each expression using the properties of logarithms. \(\log _{2} 2 a b\)

Step-by-Step Solution

Answer

\(\log_{2}2a b = 1 + \log_{2}a + \log_{2}b\)

Step 1: Apply the Product Property of Logarithms

The product property of logarithms states that the logarithm of a product is the sum of the logarithms of the factors. Therefore, we can express \( \log_{2}(2ab) \) as the sum: \[ \log_{2}(2ab) = \log_{2}2 + \log_{2}a + \log_{2}b \]

Step 2: Simplify the Logarithmic Expression

The expression can be further simplified by evaluating \( \log_{2}2 \). Since a logarithm where the base is the same as the argument equals 1, \( \log_{2}2 = 1 \). Therefore, the expression simplifies to: \[ 1 + \log_{2}a + \log_{2}b \]

Key Concepts

Product Property of Logarithms

When we talk about logarithms, one useful property to understand is the Product Property. This property helps you break down the logarithm of a product into simpler parts. It states that the logarithm of a product is the sum of the logarithms of the individual factors.
For example, if you have a logarithmic expression like \( \log_{b}(xy) \), you can use the product property to express it as \( \log_{b}x + \log_{b}y \). This makes it easier to work with, as it's often simpler to evaluate or further manipulate individual logarithms than a single, complex one.
This property is grounded in the rules of exponents. Since logarithms are actually the inverse of exponentials, if multiplication corresponds to addition in exponentials, it follows logically that the product property behaves similarly.

Logarithmic Expression Simplification

Simplifying logarithmic expressions is all about reducing the expression to its simplest form. In the context of the exercise, after applying the product property to \( \log_{2}(2ab) \), we had:

• \( \log_{2}2 + \log_{2}a + \log_{2}b \)

Next, we simplify further by recognizing that when the base and the argument of a logarithm are the same, it equals 1. Thus, \( \log_{2}2 = 1 \).
This is an important simplification property of logarithms because it makes calculations much simpler and helps in evaluating expressions quickly.
Ultimately, this simplification leads us to the expression: \( 1 + \log_{2}a + \log_{2}b \), which is cleaner and easier to interpret. Mastering these techniques is crucial for tackling more complex logarithmic problems.

Evaluating Logarithms

Evaluating logarithms involves finding the numerical value of the logarithmic expression. In simpler terms, it's about figuring out what power the base must be raised to in order to get the argument.
To evaluate logarithms like \( \log_{2}2 \), remember that the question being asked is: 2 raised to what power will give 2? The answer is 1, because 2 to the power of 1 is indeed 2. This is why \( \log_{2}2 = 1 \).

• This concept is essential when simplifying expressions because it helps you replace complex parts with simple numbers.
• Proper understanding of this concept allows you to interpret the results from simplifications correctly.

Getting comfortable with this process involves practice and remembering that logarithmic functions are the inverses of exponential functions. Once you grasp this, evaluating and working with logarithms becomes much easier.