Problem 1 - Precalculus Mathematics for Calculus

Question

\(\log x\) is the exponent to which the base 10 must be raised to get _______. So we can complete the following table for \(\log x\). $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & 10^{3} & 10^{2} & 10^{1} & 10^{0} & 10^{-1} & 10^{-2} & 10^{-3} & 10^{1 / 2} \\ \hline \log x & & & & & & & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\( \log x \) values are 3, 2, 1, 0, -1, -2, -3, \( \frac{1}{2} \).

Step 1: Understand the problem

We need to fill in the values of \( \log x \) for each value of \( x \) in the table provided. \( \log x \) indicates the exponent to which the base 10 must be raised to get \( x \).

Step 2: Calculating \( \log 10^3 \)

Since \( x = 10^3 \), the exponent to which base 10 must be raised to get \( 10^3 \) is 3. Thus, \( \log 10^3 = 3 \).

Step 3: Calculating \( \log 10^2 \)

Since \( x = 10^2 \), the exponent is 2. Thus, \( \log 10^2 = 2 \).

Step 4: Calculating \( \log 10^1 \)

Since \( x = 10^1 \), the exponent is 1. Thus, \( \log 10^1 = 1 \).

Step 5: Calculating \( \log 10^0 \)

Since \( x = 10^0 \), the exponent is 0. Thus, \( \log 10^0 = 0 \).

Step 6: Calculating \( \log 10^{-1} \)

Since \( x = 10^{-1} \), the exponent is -1. Thus, \( \log 10^{-1} = -1 \).

Step 7: Calculating \( \log 10^{-2} \)

Since \( x = 10^{-2} \), the exponent is -2. Thus, \( \log 10^{-2} = -2 \).

Step 8: Calculating \( \log 10^{-3} \)

Since \( x = 10^{-3} \), the exponent is -3. Thus, \( \log 10^{-3} = -3 \).

Step 9: Calculate \( \log 10^{1/2} \)

Since \( x = 10^{1/2} \), the exponent is \( 1/2 \). Thus, \( \log 10^{1/2} = 1/2 \).

Step 10: Fill in the table

Now that we have calculated all the values, fill in the table with \( \log x \) values:\[\begin{array}{|c|c|c|c|c|c|c|c|c|}\hlinex & 10^{3} & 10^{2} & 10^{1} & 10^{0} & 10^{-1} & 10^{-2} & 10^{-3} & 10^{1 / 2} \\hline\log x & 3 & 2 & 1 & 0 & -1 & -2 & -3 & \frac{1}{2} \\hline\end{array}\]

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication. When we see a number raised to an exponent, it indicates how many times the base number is multiplied by itself. For instance, in the expression \(10^3\), the base is 10 and the exponent is 3. This means that 10 is multiplied by itself three times (\(10 \times 10 \times 10\)), resulting in 1000.

• The expression \(a^n\) means the base \(a\) is multiplied \(n\) times.
• Exponents can be positive, like \(3\) in our earlier example, indicating multiplication.
• Negative exponents, such as \(10^{-1}\), indicate division, specifically dividing by the base raised to the positive opposite of the exponent. So \(10^{-1} = \frac{1}{10}\).

Understanding exponents is essential because it reveals the intricate relationship between numbers, especially in the context of powers of 10, critical for logarithms.

Exploring Base 10

Base 10, also known as the decimal system, is the foundation of our standard number system. It's based on ten symbols: 0 through 9, and each digit's position represents a power of 10. When we use powers of 10, we're working within the base 10 framework.

• It's called 'base 10' because it uses ten as the base for exponential notations.
• Each step to the left in a number represents an increasing power of 10, like 100 (\(10^2\)), 1,000 (\(10^3\)), etc.
• Conversely, moving right represents negative powers, such as \(0.1\) (\(10^{-1}\)), \(0.01\) (\(10^{-2}\)).

The familiarity of base 10 is precisely why it's used universally—it's simple and efficient for quick counting and calculations.

Understanding Logarithmic Functions

Logarithmic functions are the inverse operations to exponentiation. When you have an equation of the form \(y = 10^x\), its logarithmic form is \(x = \log_{10}(y)\). This highlights the key relationship: the logarithm of a number is the exponent required to raise the base to that specific number.

• A log indicates the power to which a base number must be raised to achieve a certain value.
• In base 10 logarithms (common logarithms), \(\log_{10}\) simplifies to \(\log\), without writing the base explicitly.
• For example, \(\log(1000) = 3\) because 10 raised to the power of 3 equals 1000.

Using logarithmic functions simplifies processes involving multiplication and division, making them invaluable in sciences and engineering. The transformation from exponential to logarithmic forms is a tool that provides insightful clarity into the relationships between numbers.