Problem 81 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{2} 16 $$

Step-by-Step Solution

Answer

\(\log _{2} 16 = 4\)

Step 1: Understanding the logarithm

In an equation \(\log_{b}a = n\), \(b\) is the base, \(a\) is the argument of the logarithm and \(n\) is the number the logarithm is equal to. Basically, the equation says that \(b\) raised to the power of \(n\) equals to \(a\). In our case, we are given \(\log_{2}16\), where 2 is the base and 16 is the argument.

Step 2: Using the law of exponents

We need to find a number, which when used as an exponent for the base 2, will result in 16. This could be done via trial and error or by recognizing from prior knowledge that \(2^{4} = 16\).

Step 3: Writing down the answer

The value needed to raise 2 to, in order to get the result of 16, is 4. Therefore, we can conclude that \(\log_{2}16 = 4\).

Key Concepts

Exponents: The Power Behind Numbers

An exponent is a mathematical notation that indicates the number of times a base is multiplied by itself. For instance, in the expression \(2^4\), 2 is the base and 4 is the exponent. This means that 2 is multiplied by itself four times, which equals 16. Exponents are a powerful tool in mathematics, allowing us to express large numbers succinctly and to handle calculations with repeated multiplication efficiently.

• Base: The number that is being multiplied.
• Exponent: Indicates how many times the base is used in multiplication.

Understanding exponents is crucial when dealing with logarithms, as they effectively are inverse operations. Recognizing patterns and knowing exponent rules can help simplify complex logarithmic expressions.

Base and Argument in Logarithms

In logarithmic expressions such as \(\log_b a\), the base \(b\) and argument \(a\) are crucial components to understand.

• Base: The number that is raised to a power.
• Argument: The number that results from raising the base to the given power.

For example, in the logarithm \(\log_2 16\), the 2 is the base and 16 is the argument. This expression is asking us to find the power to which 2 must be raised to yield 16. Once we've grasped these terms, we can interpret and solve logarithmic equations more confidently.

Solving Logarithmic Equations

Logarithmic equations involve finding the exponent that a base must be raised to, to achieve a given argument. Understanding this concept can simplify solving these types of problems.

• Identify the base and the argument.
• Think of the equivalent exponential equation.

For example, solving \(\log_2 16\) involves understanding that the task is to find the exponent \(n\) such that \(2^n = 16\). By recognizing that \(2^4 = 16\), we determine that \(n = 4\). Thus, \(\log_2 16 = 4\). This method of converting a logarithmic equation to its exponential form is often the most straightforward path to finding a solution.