Problem 21 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The logarithmic expression \(\log_{2} 16\) evaluates to 4.

Step 1: Understand the Problem

The problem asks us to evaluate the logarithmic expression \(\log_{2} 16\). This means we need to find the power to which the base 2 must be raised to result in 16.

Step 2: Express Logarithm in Exponential Form

Express the logarithm equation \(\log_{2} 16 = x\) in exponential form. Recall that \(\log_{b}(a) = c\) is equivalent to \(b^c = a\). Thus, the equation becomes \(2^x = 16\).

Step 3: Find Powers of 2

Find the power of 2 that equals 16. We calculate sequentially: \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\), and \(2^4 = 16\). So, \(2^4 = 16\).

Step 4: Conclude the Solution

From step 3, we found that \(x = 4\) satisfies the equation \(2^x = 16\). Therefore, \(\log_{2} 16 = 4\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a number is raised to a power or exponent. These functions look like this: \( b^x \), where \( b \) is the base and \( x \) is the exponent. The base \( b \) is a constant, and \( x \) is the variable, which tells us how many times the base multiplies by itself.

A common example of an exponential function is \( 2^x \). Here, the base is 2, and \( x \) could be any real number. Exponential functions are unique because they grow very quickly compared to linear or polynomial functions. They are often used in situations involving compound interest, population growth, or radioactive decay.

In the problem \( \log_{2} 16 = 4 \), we see the base, which is 2, raised to the power of 4 to result in 16. This demonstrates how to reverse a logarithmic operation when we convert it into an exponential form through calculation.

Logarithmic Expressions

Logarithms help us solve problems involving exponential functions by allowing us to find the exponent or power that a base must achieve to reach a particular number. An expression like \( \log_b(a) \) represents a logarithm, where \( b \) is the base, \( a \) is the number we want to reach, and the answer or result will be the exponent.

To better understand, let’s look at the expression \( \log_{2} 16 \). We are solving for the power to which 2 must be raised to get 16. This can be expressed in exponential form as \( 2^x = 16 \). By solving this, we find that \( x = 4 \), meaning \( 2^4 = 16 \). Therefore, the logarithmic expression \( \log_{2} 16 \) evaluates to 4.

Understanding logarithmic expressions can make it easier to handle real-world problems involving growth patterns and data scaling, such as sound intensity measured in decibels or the Richter scale for earthquakes.

Powers of Numbers

The concept of powers, also known as exponents, involves raising a number, known as the base, to a certain power to get another number. This power indicates how many times the base is multiplied by itself. For example, \( a^n \) represents a power, where \( a \) is the base number, and \( n \) is the exponent.

Let's simplify a common power calculation: \( 2^4 \). This means multiplying 2 by itself four times: \( 2 \times 2 \times 2 \times 2 = 16 \). Hence, the power tells us about the number of times the base factor appears in the result.

Finding powers of numbers is crucial in many areas of mathematics, including algebra and calculus, because it lets us deal effectively with both very large numbers and very small fractions. In our original problem, identifying \( 2^4 = 16 \) was key to evaluating the logarithmic expression and arriving at a correct solution.