Problem 49 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{2} \frac{1}{16}$$

Step-by-Step Solution

Answer

The simplified expression is -4.

Step 1: Understanding Logarithm Basics

To simplify the expression \( \log_{2} \frac{1}{16} \), we need to recall the basic definition of logarithms. A logarithm \( \log_b a \) answers the question: 'To what power must the base \( b \) be raised, to yield the number \( a \)?' This means \( \log_{2} \frac{1}{16} \) asks for the exponent that \( 2 \) needs to be raised to in order to get \( \frac{1}{16} \).

Step 2: Expressing Fraction as a Power of Two

The next step is to express \( \frac{1}{16} \) as a power of 2. We remember that \( 16 = 2^4 \), so \( \frac{1}{16} = 2^{-4} \) because \( 2^{-4} \) means \( 1\div2^4 \).

Step 3: Applying Logarithmic Definition

Now substitute \( \frac{1}{16} \) with \( 2^{-4} \) in the logarithm: \( \log_{2} (2^{-4}) \). The logarithm \( \log_{2}(x) = y \) implies \( 2^y = x \), so in this case, we are looking for the \( y \) such that \( 2^y = 2^{-4} \).

Step 4: Using Logarithm Properties

From the properties of logs, specifically \( \log_b(b^x) = x \), we can directly say \( \log_{2}(2^{-4}) = -4 \).

Step 5: Final Solution

Therefore, the simplified expression is \( \log_{2} \frac{1}{16} = -4 \).

Key Concepts

Logarithmic Properties

Logarithms are intriguing because they transform multiplication and division into addition and subtraction. This simplification is made possible through various logarithmic properties, which are incredibly useful when solving mathematical problems. One of the most fundamental properties is:

• \( \log_b(b^x) = x \): This states that the logarithm of a number, where the base is the same as the number being raised, is simply the exponent \( x \). For example, \( \log_2(2^3) = 3 \).
• The Change of Base Formula: Given by \( \log_b a = \frac{\log_c a}{\log_c b} \), it allows changes between different log bases, making calculations flexible.

Understanding these properties helps break down complex log expressions, as seen in the expression \( \log_2 \frac{1}{16} \), where the change from the fraction to the negative exponent allows easy simplification via the \( \log_b(b^x) \) property.

Exponentiation

Exponentiation is the process of raising a number to the power of an exponent. In simple terms, it tells us how many times to multiply a number by itself. The general form is \( b^x \), where \( b \) is the base, and \( x \) is the exponent. For example, \( 2^4 = 16 \) means multiplying 2 by itself 4 times.
Exponentiation is intimately connected with logarithms, as it is essentially the inverse operation. In the example \( \log_2 \frac{1}{16} \), we used exponentiation to express the fraction \( \frac{1}{16} \) as \( 2^{-4} \).

• Positive exponents indicate standard multiplication, like \( 3^2 = 9 \).
• Negative exponents suggest division, such as \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).

Logarithmic problems often require converting numbers into powers of a common base like this to reveal the underlying relationships and to simplify the calculations.

Simplifying Expressions

Simplifying expressions involves reducing a mathematical statement to its simplest form. This process often makes calculations more manageable and results clearer. For expressions involving logarithms, simplification usually requires leveraging properties such as those mentioned earlier. Let's break down how this was done in the exercise:

• Identifying Equivalent Forms: Recognizing that \( \frac{1}{16} \) can be rewritten as \( 2^{-4} \) is a key first step in simplification.
• Applying Properties: By using the property \( \log_b(b^x) = x \), we simplify \( \log_2(2^{-4}) \) to \(-4\).

Simplifying logarithmic expressions demands understanding of both exponential equivalents and the adept use of properties. This makes it easier to extract the exponent, which is the ultimate goal. With practice, these steps become natural, revealing the beauty and efficiency of simplified forms.