Problem 15 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The completed table is: \(f(100) = 2\) and \(f\left(\frac{1}{100}\right) = -2\).

Step 1: Understand the Function

The function given is \(f(x) = \log x\). This means for each \(x\) value, you need to calculate the logarithm to the base 10. For example, \(\log 10 = 1\), as 10 is the base to which we raise 10 to produce 10.

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

To find \(f(100)\), calculate \(\log 100\). Since \(100 = 10^2\), we have \(\log 100 = 2\). Therefore, \(f(100) = 2\).

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

To find \(f\left(\frac{1}{100}\right)\), calculate \(\log\left(\frac{1}{100}\right)\). Since \(\frac{1}{100} = 10^{-2}\), we have \(\log\left(\frac{1}{100}\right) = -2\). Therefore, \(f\left(\frac{1}{100}\right) = -2\).

Step 4: Fill in the Table

Now that you have calculated the values, fill in the table: \(f(100) = 2\) and \(f\left(\frac{1}{100}\right) = -2\).

Key Concepts

Function Table

A function table is a helpful way to organize the output of a mathematical function given specific input values. By laying out values of the independent variable, in this case, \(x\), alongside their corresponding outputs, this table helps visualize and keep track of how the inputs and outputs relate.

For our exercise, we have the function \(f(x) = \log x\). This requires us to evaluate the logarithms of various \(x\) values. The table for this function pairs each input \(x\) with its calculated output \(f(x)\).

Steps to use a function table for logarithmic functions:

• Identify the function you are dealing with. Here, it's \(f(x) = \log x\).
• Determine the input values you want to use. Our inputs are \(100\) and \(\frac{1}{100}\).
• Compute the function output for each input using the logarithmic calculation.
• Fill in the table with these outputs accordingly.

This approach helps you easily manage calculations and track them in an organized manner.

Logarithm Base 10

The logarithm base 10, also known as the common logarithm, is a mathematical operation used to determine the power which 10 must be raised to to obtain a specific number. In notation, it is usually written as \(\log_{10}\), or just \(\log\) when the base is omitted, since base 10 is so common.

Key Points about Logarithm Base 10:

• It's frequently used in scientific calculations as a way to measure exponential growth or decay.
• For example, \(\log_{10} 100\) calculates to \(2\) because \(10^2 = 100\).
• Logarithms convert multiplicative relationships into additive ones, which simplifies many calculations.

By understanding this base, you can simplify complex multiplication processes and make calculations more manageable.

Evaluate Logarithms

Evaluating logarithms involves determining the power to which a base number, often 10, must be raised in order to result in a given number. Our task is to evaluate specific logarithmic values which involve these calculations.

When evaluating logarithms:

• Think of the logarithm as the opposite of exponentiation. It answers, "To what power, 10 must be raised, to produce this number?"
• For instance, to evaluate \(\log 100\), recognize that \(100 = 10^2\), hence \(\log 100 = 2\).
• Similarly, for \(\log \frac{1}{100}\), recognize that \(\frac{1}{100} = 10^{-2}\), resulting in \(\log \frac{1}{100} = -2\).

This step-by-step process simplifies the evaluation of logarithms, making them easier to understand and apply in different scenarios.