Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the average rate of change \(f(x)=\log _{2} x\) from 4 to \(16 .\)

Short answer

\frac{1}{6}

Step-by-step solution

Identify the function and interval

The function given is \( f(x) = \log_{2}{x} \). The interval over which to find the average rate of change is from \ x = 4 \ to \ x = 16 \.

Apply the average rate of change formula

The average rate of change of a function \( f(x) \) from \( a \text{ to } b \) is given by \ \frac {f(b) - f(a)}{b - a} \.

Evaluate the function at the endpoints

Compute \( f(4) \) and \( f(16): \ f(4) = \log_{2}{4} = 2 \ (since \ 2^{2} = 4 \), and \ f(16) = \log_{2}{16} = 4 \ (since \ 2^{4} = 16).

Substitute and simplify

Substitute the values \( f(16) \) and \( f(4) \) into the average rate of change formula: \ \frac {f(16) - f(4)}{16 - 4} = \ \frac {4 - 2}{16 - 4} = \frac {2}{12} = \frac{1}{6}.

Key concepts

Logarithmic Functions

Logarithmic functions are a fundamental part of algebra and calculus. They are the inverse of exponential functions. For instance, if you have an exponential function like \(2^x = y\), the corresponding logarithm would be written as \(x = \log_{2}(y)\). This means that logarithms help us find the exponent given the base and the result. In our given problem, we deal with \(f(x) = \log_{2}(x)\), where the base is 2. Logarithms have several properties that make them useful:
\textbf{1. Change of Base Formula:} To change bases of logarithms:\[\frac{\log_{b}(a)}{\log_{c}(a)}\]\ The new base is \(c\).\
\textbf{2. Logarithm of a Product:} When multiplying, the logarithms add:\[\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\]\
\textbf{3. Logarithm of a Power:} When taking the log of a power, the exponent comes down:\[\frac{\log_{b}(x^y)}{y}\]\
Understanding these properties can clarify how we manipulate logarithmic expressions efficiently. 

Rate of Change

In mathematics, the rate of change measures how one quantity varies concerning another. This is crucial for understanding and interpreting data, functions, and graphs.
For example, the average rate of change of a function \(f(x)\) over an interval \[a, b\]\ is given by:\[\frac{f(b) - f(a)}{b - a}\]\ This formula tells us the average amount \(f(x)\) changes per unit across the interval. Calculating the rate of change involves:

• Identifying the endpoints \(a\) and \(b\).
• Computing the function \(f(x)\) values at these endpoints.
• Substituting these values into the formula.

For our problem, we used \(f(x) = \log_{2}(x)\) with \(x = 4 \text{to } 16\). By substituting, we obtained: \[\frac{f(16) - f(4)}{16 - 4} = \frac{4 - 2}{12} = \frac{2}{12} = \frac{1}{6}\]\ documenting an average rate of change of \(\frac{1}{6}\)\.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It includes everything from solving simple equations to more complex problems like systems of equations and functions. Algebra forms the foundation for many advanced topics in mathematics.


The given problem uses several core algebraic concepts:

• ###Calculating values of a function: This is done by substituting the values into the function \(f(x) = \log_{2}(x)\).
• ###Understanding intervals: The interval \[4, 16\]\ is where we evaluate the function.
• ###Using the difference quotient: This is a form of slope formula essential in calculus, showing the average rate of change across the interval.

By grasping these foundational algebra principles, students can effectively navigate logarithmic functions and rate of change problems, which is vital for mastering higher-level math concepts.