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Chapter 3: Problem 10

Suppose \(k\) is a positive integer such that \(\log k \approx 83.2\). How many digits does \(k\) have?

Short Answer

Expert verified
The number of digits in \(k\) is 84.

Step by step solution

01

Rewrite the logarithm expression

The given logarithm is \(\log k \approx 83.2\). Since it is a common logarithm (base 10), we can rewrite it as: \[k \approx 10^{83.2}\]
02

Find the range of values \(k\) could take

To find the number of digits in \(k\), let's determine between which two integers \(k\) could lie. We know that \(k\) is somewhere between \(10^{83}\) and \(10^{84}\), as when you increase the exponent by 1, the number of digits increases by 1. \[10^{83} \le k < 10^{84}\]
03

Determine the number of digits in \(k\)

All numbers in the above-defined range will have the same number of digits, as numbers from \(10^{83}\) to \(10^{84}-1\) will have 84 digits. So, the value of \(k\) has 84 digits.

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