What is the number of digits in 7 25 , 8 23 and 9 20 respectively? [Given log 10 2 = 0.301, log 10 3 = 0.477, log 10 7 = 0.845]
22, 21, 20
Finding the Number of Digits Using Logarithms
To find the number of digits in a large positive integer \(N\), we can use the concept of common logarithms (logarithms base 10). The number of digits in \(N\) is given by the formula:
Number of digits \(= \lfloor \log_{10} N \rfloor + 1\)
Here, \(\lfloor \log_{10} N \rfloor\) represents the characteristic of the common logarithm of \(N\), which is the greatest integer less than or equal to \(\log_{10} N\).
Calculating the Number of Digits in 725
We need to find the number of digits in \(7^{25}\). Using the formula, we first calculate \(\log_{10}(7^{25})\).
Using the logarithm property \(\log_b (a^c) = c \log_b a\):
\(\log_{10}(7^{25}) = 25 \log_{10} 7\)
We are given \(\log_{10} 7 = 0.845\). Substituting this value:
\(\log_{10}(7^{25}) = 25 \times 0.845\)
\(\log_{10}(7^{25}) = 21.125\)
The characteristic of \(\log_{10}(7^{25})\) is the integer part, which is \(\lfloor 21.125 \rfloor = 21\).
The number of digits in \(7^{25}\) is the characteristic plus 1:
Number of digits in \(7^{25} = 21 + 1 = 22\).
Calculating the Number of Digits in 823
We need to find the number of digits in \(8^{23}\). We first calculate \(\log_{10}(8^{23})\).
Since \(8 = 2^3\), we can write \(8^{23}\) as \((2^3)^{23} = 2^{3 \times 23} = 2^{69}\).
So, we need to find \(\log_{10}(2^{69})\). Using the logarithm property \(\log_b (a^c) = c \log_b a\):
\(\log_{10}(8^{23}) = \log_{10}(2^{69}) = 69 \log_{10} 2\)
We are given \(\log_{10} 2 = 0.301\). Substituting this value:
\(\log_{10}(8^{23}) = 69 \times 0.301\)
\(\log_{10}(8^{23}) = 20.769\)
The characteristic of \(\log_{10}(8^{23})\) is the integer part, which is \(\lfloor 20.769 \rfloor = 20\).
The number of digits in \(8^{23}\) is the characteristic plus 1:
Number of digits in \(8^{23} = 20 + 1 = 21\).
Calculating the Number of Digits in 920
We need to find the number of digits in \(9^{20}\). We first calculate \(\log_{10}(9^{20})\).
Since \(9 = 3^2\), we can write \(9^{20}\) as \((3^2)^{20} = 3^{2 \times 20} = 3^{40}\).
So, we need to find \(\log_{10}(3^{40})\). Using the logarithm property \(\log_b (a^c) = c \log_b a\):
\(\log_{10}(9^{20}) = \log_{10}(3^{40}) = 40 \log_{10} 3\)
We are given \(\log_{10} 3 = 0.477\). Substituting this value:
\(\log_{10}(9^{20}) = 40 \times 0.477\)
\(\log_{10}(9^{20}) = 19.080\)
The characteristic of \(\log_{10}(9^{20})\) is the integer part, which is \(\lfloor 19.080 \rfloor = 19\).
The number of digits in \(9^{20}\) is the characteristic plus 1:
Number of digits in \(9^{20} = 19 + 1 = 20\).
Summary of Number of Digits
Based on our calculations using logarithms:
| Number | Logarithm (base 10) | Characteristic (\(\lfloor \text{log}_{10} N \rfloor\)) | Number of Digits (Characteristic + 1) |
|---|---|---|---|
| \(7^{25}\) | \(21.125\) | \(21\) | \(22\) |
| \(8^{23}\) | \(20.769\) | \(20\) | \(21\) |
| \(9^{20}\) | \(19.080\) | \(19\) | \(20\) |
Thus, the number of digits in \(7^{25}\), \(8^{23}\), and \(9^{20}\) are 22, 21, and 20 respectively.
Revision Table: Logarithms and Number of Digits
| Concept | Description |
|---|---|
| Common Logarithm | Logarithm with base 10, denoted as \(\log_{10} x\). |
| Characteristic | The integer part of the common logarithm of a number \(N\). For \(N \ge 1\), it is \(\lfloor \log_{10} N \rfloor\). |
| Number of Digits | For a positive integer \(N\), the number of digits is equal to its characteristic plus one (\(\lfloor \log_{10} N \rfloor + 1\)). |
| Logarithm Property | \(\log_b (a^c) = c \log_b a\) |
Additional Information: Applications of Logarithms
Logarithms are not just useful for finding the number of digits in large numbers. They have many applications in various fields:
- Science: Used in scales like the Richter scale for earthquakes (log base 10), pH scale for acidity (log base 10), and decibel scale for sound intensity (log base 10).
- Mathematics: Solving exponential equations, simplifying calculations, calculus, and complexity analysis in algorithms.
- Finance: Calculating compound interest and growth rates.
- Computer Science: Analyzing the efficiency of algorithms (e.g., logarithmic time complexity).
The common logarithm (base 10) is particularly useful because our number system is base 10. This makes the characteristic directly related to the magnitude or number of digits of a number. The mantissa (the fractional part of the logarithm) helps determine the sequence of digits in the number.
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