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Chapter 7: Problem 11

Find the smallest integer \(n\) such that \(0.8^{n}<10^{-100}\).

Short Answer

Expert verified
The smallest integer \(n\) such that \(0.8^n < 10^{-100}\) is 133.

Step by step solution

01

Write the Inequality in terms of Logarithm

First write the given inequality in terms of logarithm: \(0.8^n < 10^{-100}\) Taking the logarithm of both sides to the base 10, we get: \(n\log_{10}(0.8) < -100\)
02

Use the Properties of Logarithm

Use the properties of logarithm to simplify the inequality. We know that \(\log_{10}(a^b)=b\log_{10}(a)\), and in our case we have: \(n\log_{10}(0.8) < -100\)
03

Solve for n

Now, solve for n by dividing both sides of the inequality by \(\log_{10}(0.8)\): \(n < \frac{-100}{\log_{10}(0.8)}\)
04

Calculate the Value

Calculate the value of the right side: \(n < \frac{-100}{\log_{10}(0.8)} \approx 132.2877\)
05

Find the Smallest Integer n

As n should be an integer, and the inequality is strict (n can't be equal to the right side), we need to find the smallest integer greater than 132.2877, so: \(n=133\) The smallest integer n such that \(0.8^n < 10^{-100}\) is 133.

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