Problem 95 - Precalculus Mathematics for Calculus

Question

Compare log 1000 to the number of digits in 1000 . Do the same for 10,000 . How many digits does any number between 1000 and 10,000 have? Between what two values must the common logarithm of such a number lie? Use your observations to explain why the number of digits in any positive integer \(x\) is \(\|\log x\|+1 .\) (The symbol \(\|n\|\) is the greatest integer function defined in Section 2.2 . ) How many digits does the number \(2^{100}\) have?

Step-by-Step Solution

Answer

\(2^{100}\) has 31 digits.

Step 1: Understand the Problem

We need to compare the common logarithm (base 10 logarithm) of the number 1000 and 10,000 to the number of digits in those numbers. Then, we explore these patterns for numbers between 1000 and 10,000 and interpret why the number of digits in a number is given by \( \lfloor \log x \rfloor + 1 \), where \( \lfloor n \rfloor \) is the greatest integer function.

Step 2: Calculate the Logarithm and Digits for 1000

The number 1000 has 4 digits. Its logarithm is calculated as \( \log_{10} 1000 = 3 \) because \( 10^3 = 1000 \). The greatest integer less than or equal to 3 is 3, thus \( \lfloor 3 \rfloor + 1 = 4 \) matches the number of digits.

Step 3: Calculate the Logarithm and Digits for 10,000

The number 10,000 has 5 digits. Its logarithm is \( \log_{10} 10,000 = 4 \) because \( 10^4 = 10,000 \). The greatest integer less than or equal to 4 is 4, so \( \lfloor 4 \rfloor + 1 = 5 \) matches the number of digits.

Step 4: Generalize for Numbers Between 1000 and 10,000

A number between 1000 and 10,000 will always have 4 digits. For numbers \( n \) in this range, \( 3 \leq \log_{10} n < 4 \). Therefore, \( \lfloor \log_{10} n \rfloor = 3 \), giving us \( \lfloor \log_{10} n \rfloor + 1 = 4 \), the correct count of digits.

Step 5: Why \( \lfloor \log x \rfloor + 1 \) Matches the Digits

A number \( x \) has \( d \) digits when \( 10^{d-1} \leq x < 10^d \). Taking logarithms gives \( d-1 \leq \log_{10} x < d \). Thus, \( \lfloor \log_{10} x \rfloor = d-1 \) and \( \lfloor \log_{10} x \rfloor + 1 = d \), matching the number of digits in \( x \).

Step 6: Determine the Number of Digits in \( 2^{100} \)

First, calculate the logarithm base 10 of \( 2^{100} \): \( \log_{10} (2^{100}) = 100 \cdot \log_{10} 2 \approx 100 \cdot 0.3010 = 30.1 \). Then \( \lfloor 30.1 \rfloor + 1 = 31 \). Therefore, \( 2^{100} \) has 31 digits.

Key Concepts

Common Logarithm

A common logarithm is a logarithm with base 10. It is denoted by \( \log_{10} \), though it's often simply written as \( \log \). Common logarithms are useful because they relate to the number of zeroes that follow the digit 1 when a number is expressed in standard form using powers of 10. In our example, \( \log_{10} 1000 = 3 \) since \( 10^3 = 1000 \). This indicates that the number 1000 is 10 raised to the third power. Understanding this concept helps us interpret how many digits a number has when using powers of 10. If a number \( x \) can be expressed as \( 10^d \), the result from the logarithm shows how closely \( x \) aligns to a power of ten.

Greatest Integer Function

The greatest integer function, also known as the floor function, is a mathematical function that produces the largest integer less than or equal to a given number. It is denoted as \( \lfloor n \rfloor \). For example, \( \lfloor 4.7 \rfloor \) equals 4, since 4 is the greatest integer less than 4.7.
\( \lfloor \log_{10} x \rfloor \) helps determine the number of digits in a number. If we look at \( \log_{10} 1000 = 3 \), \( \lfloor 3 \rfloor \) is 3, showing the number starts with a power of 10 related to the number of digits. Adding 1 gives us the total digit count, because any whole number between \( 10^3 \) and \( 10^4 \) has 4 digits.

Number of Digits

Counting the number of digits in a number is essential for understanding its size and scale. The formula \( \lfloor \log_{10} x \rfloor + 1 \) helps count digits efficiently. For any positive integer \( x \), by seeing where it falls between powers of ten, we can ascertain its digits.
Let's say \( x = 5000 \), and we find \( \log_{10} 5000 \approx 3.6989 \). Taking the floor gives \( \lfloor 3.6989 \rfloor = 3 \), and when we add 1, we conclude it has 4 digits. This method applies broadly, even to numbers expressed as powers, such as \( 2^{100} \), which we calculated has 31 digits through \( \log_{10} (2^{100}) \approx 30.1 \). Each step confirms how the logarithmic relationship plays a pivotal role in understanding numerical magnitude.