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Chapter 3: Problem 14

Suppose \(M\) is a positive integer such that \(\log M \approx 50.3 .\) How many digits does \(M^{4}\) have?

Short Answer

Expert verified
The number of digits in \(M^4\) is 202.

Step by step solution

01

Find the value of \(\log (M^4)\)

Using the properties of logarithms, we can find the value of \(\log (M^4)\) as follows: \(\log (M^4) = 4 \times \log M\) Since we are given \(\log M \approx 50.3\), we can plug this value into the equation to get: \(\log (M^4) = 4 \times 50.3 \approx 201.2\)
02

Determine the number of digits for powers of 10

To determine the number of digits from its logarithm value, we can identify the two closest powers of 10. From the value of \(\log (M^4) = 201.2\), we know that \(10^{201} \leq M^4\) and \(M^4 < 10^{202}\).
03

Finding the number of digits in \(M^4\)

The smallest number with \(k\) digits is \(10^{k-1}\) and the largest number with \(k\) digits is \(10^k - 1\). From step 2, we know that \(10^{201} \leq M^4 < 10^{202}\), which means \(M^4\) has 202 digits. The number of digits in \(M^4\) is 202.

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