Problem 96 - Precalculus : Building Concepts and Connections

Question

Explain how you could use the following table of values for the function \(f(x)=10^{x}\) to find the given quantity. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$\log 3$$

Step-by-Step Solution

Answer

\(\log 3 = 0.4771\)

Step 1: Understand Logarithms

Firstly, we need to understand that \(\log_b a\) is the exponent to which 'b' must be raised to get 'a'. So, to solve \(\log 3\), we want to know what power we must raise '10' to, to get '3'.

Step 2: Consult the Table

Next we consult the table for this power. Find a row where \(f(x) = 10^x = 3\). 'x' in that row will be the answer to our problem because that's the exponent 'x' such that \(10^x = 3\).

Step 3: Find the Answer

Upon consulting the table, we can see that 'x' equals '0.4771' when \(f(x) = 10^x = 3\). Therefore, \(\log 3 = 0.4771\).

Key Concepts

Exponentiation

Exponentiation is an essential mathematical operation that involves raising a number, known as the base, to the power of an exponent. When you see an expression like \(10^x\), it represents 10 being multiplied by itself \(x\) times. Exponentiation allows us to scale numbers rapidly, especially when dealing with large values or growth calculations. Understanding this operation is crucial as it is a cornerstone in mathematics used in various applications ranging from computing interest rates to solving exponential growth problems.

Exponentiation is directly linked with logarithms. The concept can be thought of as the reverse of taking a logarithm. For example, if you know that \(10^x = 3\), the problem asks for the logarithm, essentially solving for \(x\) using the logarithm function, denoted as \(\log_{10} 3\). This ties exponentiation closely to logarithms. The process of finding the exponent that achieves a specific growth, like multiplying to a certain number, parallels understanding exponential growth itself. Recognizing how exponentiation functions helps elucidate the relationship between numbers in a logarithmic context.

Function Tables

Function tables are helpful tools in mathematics that display input-output pairs for particular functions. These tables provide a structured way to understand the relationship between variables. The table for the function \(f(x)=10^x\) lists values of \(x\) alongside the corresponding outputs of \(10^x\). Such tables allow us to quickly see how the function behaves for different input values.

In the context of this problem, the function table plays a pivotal role in finding the logarithm of a number. By consulting the table, we can efficiently determine the value of \(x\) that results in a desired function output. This is particularly crucial for logarithms, where we need to find the exponent that satisfies an equation like \(10^x = 3\). With the help of function tables, such tasks become straightforward as they display the precise input-output pairs directly,

• Easy lookup of function values
• Understanding input-output relationships quickly
• Serving as a reference for finding exponents or logarithms

Essentially, a well-constructed function table becomes an invaluable resource for solving exponential and logarithmic equations.

Base 10

Base 10, often known as the decimal system, is one of the most common bases used in mathematics and everyday life. It hinges on the number 10, utilizing digits from 0 to 9. When we talk about the base in mathematics, it's the number which is raised to various powers in expressions involving exponents and logarithms.

In our context, the term "base 10" is important because logarithms are often calculated to this base, especially in scientific and engineering fields. For a logarithm such as \(\log_{10} 3\), base 10 tells us that we are finding the power to which 10 needs to be raised to yield the number 3.

• Understanding the base clarifies the logarithmic calculations
• Base 10 is crucial for standard logarithmic functions
• A fundamental system for numerical calculations

Knowing that base 10 is the foundation of the logarithm in this question, it simplifies how we approach and solve such logarithmic problems by directly relating them to base 10 exponentiation.