Problem 34 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \log _{2} \frac{1}{4} $$

Step-by-Step Solution

Answer

\( \log_{2} \frac{1}{4} = -2 \).

Step 1: Recognize the Logarithmic Property

The expression is \( \log_{2} \frac{1}{4} \). We know that for a logarithmic function, \( \log_{b}(a) = c \) means that \( b^{c} = a \). Here, our base \( b = 2 \) and \( a = \frac{1}{4} \). We are tasked with finding the exponent \( c \).

Step 2: Express \( \frac{1}{4} \) as a Power of 2

Observe that \( \frac{1}{4} \) can be expressed as a power of 2. Note that \( 4 = 2^{2} \), hence \( \frac{1}{4} = 2^{-2} \).

Step 3: Apply Logarithmic Identity

Using the identity \( \log_{b}(b^{c}) = c \), substitute \( 2^{-2} \) back into the logarithmic expression: \[\log_{2}(2^{-2}) = -2\] Thus, the logarithm simplifies to \( -2 \).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is essential when simplifying logarithmic expressions. Logarithms allow us to work backward from exponentiation. If you have an expression like \( \log_b(a) \), this means you are looking for the power \( c \) such that the base \( b \) raised to the power \( c \) equals \( a \). Logarithmic properties are rules that help simplify such expressions easily. Here are some key properties:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \).
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \).
• Power Rule: \( \log_b(M^k) = k\log_b(M) \).
• Base Change Rule: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), for any positive base\( k \).

In our example, the identity \( \log_b(b^c) = c \) is used, which makes solving our problem straightforward once \( a \) is expressed in terms of \( b \). The log property states that if the argument of the log and the base are the same, the log reduces to the exponent.

Power of a Number

Understanding how to express numbers as powers is a valuable skill. The expression \( \frac{1}{4} \) set against a base 2 provides a great insight into this concept. Knowing how numbers can be rewritten as powers of another number simplifies many logarithmic expressions. We know that 4 can be broken down into \( 4 = 2^2 \). Therefore, \( \frac{1}{4} \) can be rewritten as \( \frac{1}{2^2} \), which equals \( 2^{-2} \). This transformation is crucial. By expressing a fraction as a negative exponent, it becomes straightforward to apply logarithmic identities. When handling fractional values in logarithmic problems, always look for a base that capitalizes on the relationship between the base of the logarithm and the numerator or denominator. This will simplify solving apploringly complex-looking expressions down to basics.Knowing these transformations and identities not only helps in logarithmic expressions, but is a fundamental mathematical skill that crosses into algebra and higher-level mathematics.

Logarithmic Expression

A logarithmic expression is any mathematical expression involving a logarithm. The goal often is to evaluate, simplify, or solve the expression. In our problem, we are tasked with simplifying \( \log_2\left(\frac{1}{4}\right) \). Understanding logarithmic expressions often involves rewriting them using known identities and properties, as discussed in prior sections. A crucial step here involves recognizing when an expression can be rewritten with a common base, such as in our exercise. In our case:

• Realizing that \( \frac{1}{4} = 2^{-2} \) transformed the log base into the same form, allowing for direct application of the logarithm identity \( \log_b(b^c) = c \).
• This resulted in simply extracting the exponent: \( \log_2(2^{-2}) = -2 \).

By expressing the fraction appropriately, we reduced the seemingly complex problem into a basic logarithmic calculation. This simplification demonstrates how recognizing patterns and properties in logarithmic expressions can significantly aid in solving them quickly.