Problem 15 - Intermediate Algebra

Question

Complete the table of values. $$ f(x)=\log x $$ $$ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 100 & \\ \hline \frac{1}{100} & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

For \( x = 100, f(x) = 2 \) and for \( x = \frac{1}{100}, f(x) = -2 \).

Step 1: Understand the Problem

The exercise asks us to find the logarithm of two numbers, 100 and \( \frac{1}{100} \), using the function \( f(x) = \log x \), and then complete the table with these values.

Step 2: Calculate \( f(100) \)

We need to find \( \log 100 \). Since \( 100 = 10^2 \), we use the logarithm property \( \log(a^b) = b \log a \). Here, \( \log 10 = 1 \), so \( \log 100 = 2 \times \log 10 = 2 \). Therefore, \( f(100) = 2 \).

Step 3: Calculate \( f \left( \frac{1}{100} \right) \)

We need to find \( \log \left( \frac{1}{100} \right) \). Using the property \( \log \left( \frac{1}{a} \right) = -\log a \), where \( a = 100 \), we find \( \log \left( \frac{1}{100} \right) = -\log 100 = -2 \). Hence, \( f \left( \frac{1}{100} \right) = -2 \).

Step 4: Complete the Table

Now that we have found \( f(100) = 2 \) and \( f \left( \frac{1}{100} \right) = -2 \), we can fill in the table: \[\begin{array}{|c|c|}\hline x & f(x) \\hline 100 & 2 \\hline \frac{1}{100} & -2 \\hline\end{array}\]

Key Concepts

Properties of Logarithms

Logarithms come with several helpful properties that simplify their computation. These properties are not just mathematical tricks; they form the backbone of solving complex logarithmic functions.

• The Power Rule: One of the key properties is the power rule, expressed as \( \log(a^b) = b \log a \). This rule helps to simplify expressions involving exponents. For example, when computing \( \log 100 \), recognize that \( 100 = 10^2 \). Hence, \( \log 100 = 2 \times \log 10 \), making the calculation straightforward because \( \log 10 = 1 \).
• The Quotient Rule: Another useful property is the quotient rule. It states that \( \log \left( \frac{1}{a} \right) = -\log a \). This rule helps in finding the logarithm of fractions. For instance, \( \log \left( \frac{1}{100} \right) \) can be determined by realizing it as \( -\log 100 \), which is \(-2\).

These properties are essential building blocks when working with logarithms, making it easier to break down and understand complex calculations.

Function Evaluation

Evaluating a function means calculating its output for a specific input. In this context, we are dealing with the logarithmic function \( f(x) = \log x \). This function tells us the power to which a base, typically 10, must be raised to obtain the value \( x \).
To practice function evaluation, let's consider two cases:

• For \( x = 100 \): Recognize that \( 100 = 10^2 \). Using the power rule of logarithms, \( \log 100 = 2 \times \log 10 \). Since \( \log 10 = 1 \), it follows that \( f(100) = 2 \).
• For \( x = \frac{1}{100} \): Using the quotient rule, \( \log \left( \frac{1}{100} \right) = -\log 100 = -2 \). So, \( f \left( \frac{1}{100} \right) = -2 \).

Understanding these steps equips you with the ability to perform function evaluations on any logarithmic expression.

Logarithm Table

A logarithm table is a handy reference to quickly find logarithmic values without calculating them each time. Historically, these tables were extensively used before the advent of calculators.
Logarithm tables typically list values of \( \log x \) for a range of \( x \). These could be for different bases, but base 10—known as the common logarithm—is often used.

• Using the Table: When faced with the task of finding \( \log 100 \), you would refer to a logarithm table if it lists base 10 values. It provides a quick look-up to know that \( \log 100 = 2 \).
• Advanced Computations: For more complex inputs, the table becomes especially valuable. By showing values for fractions and complex numbers, it aids in accurate computation without new calculations.

While digital tools have largely replaced them, understanding how these tables are built and used can deepen your understanding of logarithms.