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Chapter 4: Problem 1

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

Short Answer

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

Step by step solution

01

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 \( x \).
02

Calculating \( \log 10^3 \)

Since \( x = 10^3 \), the exponent to which base 10 must be raised to get \( 10^3 \) is 3. Thus, \( \log 10^3 = 3 \).
03

Calculating \( \log 10^2 \)

Since \( x = 10^2 \), the exponent is 2. Thus, \( \log 10^2 = 2 \).
04

Calculating \( \log 10^1 \)

Since \( x = 10^1 \), the exponent is 1. Thus, \( \log 10^1 = 1 \).
05

Calculating \( \log 10^0 \)

Since \( x = 10^0 \), the exponent is 0. Thus, \( \log 10^0 = 0 \).
06

Calculating \( \log 10^{-1} \)

Since \( x = 10^{-1} \), the exponent is -1. Thus, \( \log 10^{-1} = -1 \).
07

Calculating \( \log 10^{-2} \)

Since \( x = 10^{-2} \), the exponent is -2. Thus, \( \log 10^{-2} = -2 \).
08

Calculating \( \log 10^{-3} \)

Since \( x = 10^{-3} \), the exponent is -3. Thus, \( \log 10^{-3} = -3 \).
09

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

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

Fill in the table

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攊t'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.

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Most popular questions from this chapter

What is wrong with the following argument? $$\begin{aligned}\log 0.1 &<2 \log 0.1 \\\&=\log (0.1)^{2} \\\&=\log 0.01 \\\\\log 0.1 &<\log 0.01 \\ 0.1 &<0.01\end{aligned}$$

Find the domain of the function. $$h(x)=\sqrt{x-2}-\log _{5}(10-x)$$

A spectrophotometer measures the concentration of a sample dissolved in water by shining a light through it and recording the amount of light that emerges. In other words, if we know the amount of light that is absorbed, we can calculate the concentration of the sample. For a certain substance the concentration (in moles per liter) is found by using the formula $$C=-2500 \ln \left(\frac{I}{I_{0}}\right)$$ where \(I_{0}\) is the intensity of the incident light and \(I\) is the intensity of light that emerges. Find the concentration of the substance if the intensity \(I\) is \(70 \%\) of \(I_{0}\).

Growth of an Exponential Function Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? (a) One million dollars at the end of the month (b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, \(2^{n}\) cents on the \(n\) th day

Solve for \(x: 2^{2 / \log _{5} x}=\frac{1}{16}\)
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