Problem 35 - Algebra 2

Question

Evaluate each expression. \(\log _{2} 16\)

Step-by-Step Solution

Answer

\( \log_{2} 16 = 4 \)

Step 1: Understand the Logarithmic Expression

The expression \( \log_{2} 16 \) asks for the power to which the base 2 must be raised to obtain 16. In other words, we need to find \( x \) such that \( 2^x = 16 \).

Step 2: Express 16 as a Power of 2

To evaluate \( \log_{2} 16 \), it can be helpful to express 16 in terms of base 2. Notice that \( 16 = 2^4 \).

Step 3: Apply the Definition of Logarithm

Set up the equation using the definition of logarithm: \( \log_{2} 16 = \log_{2} (2^4) \). By the properties of logarithms, we know that \( \log_{b} (b^x) = x \).

Step 4: Calculate the Logarithm

Using the property from Step 3, \( \log_{2} (2^4) = 4 \). Therefore, \( \log_{2} 16 = 4 \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements used to determine the exponent needed for a specific base to reach a certain number. In the expression \( \log_{b} a \), "\( b \)" is the base and "\( a \)" is the number we want to obtain by raising the base to some power. Essentially, the expression is asking, "What power must we raise \( b \) to, in order to get \( a \)?" The result of this expression is the exponent.For example, in the expression \( \log_{2} 16 \), we are determining the exponent to which 2 must be raised to reach 16. It's like solving a small puzzle using exponents. Understanding logarithmic expressions is foundational to evaluating and manipulating them using different properties of logarithms.

Properties of Logarithms

Properties of logarithms provide rules and shortcuts to solve logarithmic expressions more easily. One important property is the Power Rule: \( \log_{b} (b^x) = x \). This tells us that if the number we're taking the logarithm of is a power of the base, the result is simply the exponent.There are also other key properties, such as:

• The Product Rule: \( \log_{b} (MN) = \log_{b} M + \log_{b} N \)


• The Quotient Rule: \( \log_{b} \left( \frac{M}{N} \right) = \log_{b} M - \log_{b} N \)


• The Change of Base Formula: \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \)

These properties simplify complex expressions and allow us to evaluate logarithms using different approaches. When evaluating the expression \( \log_{2} 16 \), the Power Rule is directly applicable because 16 is a power of 2: 16 equals \( 2^4 \). Thus, \( \log_{2} 16 = 4 \).

Evaluating Logarithms

Evaluating logarithms involves finding the exponent that a particular base must be raised to in order to produce a given number. Here's a straightforward process to follow:1. **Rewrite the Number as a Power of the Base**: Like in \( \log_{2} 16 \), express 16 as a power of the base 2, which is \( 2^4 \).2. **Apply the Logarithmic Properties**: Use the property \( \log_{b} (b^x) = x \) to recognize that \( \log_{2} (2^4) = 4 \).3. **Simplify the Expression**: With the properties applied correctly, simplify the expression to get the result.By using these steps, evaluating logarithmic expressions becomes an easier task. Recognizing and expressing numbers as powers of a base is a crucial skill when working with logarithms. This process is not only applicable to base 2 but to any base you encounter while working through logarithmic problems.