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Chapter 3: Problem 51

Find the smallest integer \(M\) such that \(5^{1 / M}<1.01\).

Short Answer

Expert verified
The smallest integer \(M\) such that \(5^{1/M}<1.01\) is \(M=73\).

Step by step solution

01

Take the logarithm of both sides of the inequality

We'll take the natural logarithm (ln) of both sides of the inequality, which doesn't change the inequality's direction since ln is an increasing function. So we get: \[\ln(5^{1/M}) < \ln(1.01)\]
02

Use the power rule of logarithms

The power rule states that \(\ln(a^b) = b\ln(a)\). Applying this rule to our inequality, we get: \[\frac{1}{M} \ln(5) < \ln(1.01)\]
03

Isolate the variable M

To isolate the variable \(M\), we first divide by \(\ln(5)\): \[\frac{1}{M} < \frac{\ln(1.01)}{\ln(5)}\] Next, take the reciprocal of both sides: \[M > \frac{\ln(5)}{\ln(1.01)}\]
04

Calculate the value of the inequality

Now, we need to find the smallest integer \(M\) that satisfies the inequality. First, compute the value of the inequality: \[M > \frac{\ln(5)}{\ln(1.01)} \approx 72.47\]
05

Identify the smallest integer value

Since \(M\) must be an integer and \(M > 72.47\), the smallest integer value for \(M\) is 73. So, the smallest integer \(M\) such that \(5^{1/M}<1.01\) is \(M=73\).

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