Problem 12 Find the smallest integer $n$ ... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 7: Problem 12

Find the smallest integer $n$ such that $0.9^{n}<10^{-200}$.

Short Answer

Expert verified

The smallest integer n that satisfies the inequality $0.9^n < 10^{-200}$ is n = 4371.

Step by step solution

Rewrite the inequality with the same base and apply the logarithm

On the left side, we can rewrite 0.9 as $9 \cdot 10^{-1}$ and the right side is already in the desired form. So, the inequality becomes: $(9 \cdot 10^{-1})^n < 10^{-200}$ Now, we will take a logarithm on both sides. Let's use log base 10 since that will make the calculations a little easier: \[\log_{10} ((9 \cdot 10^{-1})^n) < \log_{10}(10^{-200})\]

Apply properties of logarithms and solve for n

Apply the power property of logarithms, which is $\log_a(b^c) = c \log_a b$: $n \log_{10} (9 \cdot 10^{-1}) < -200$ Now, we use the product property of logarithms, which is $\log_a(bc) = \log_a b + \log_a c$: $n (\log_{10}(9) + \log_{10}(10^{-1})) < -200$ Simplify the equation by calculating the logarithm values: $n (0.95424 - 1) < -200$ Now we have a simple linear inequality to solve for n: $-0.04576n < -200$ Divide both sides by -0.04576 and reverse the inequality sign $n > \frac{-200}{-0.04576} $ Calculate the result: $n > 4370.97$

Find the smallest integer n

Since n must be an integer, the smallest integer value of n that satisfies the inequality is 4371. So, the smallest integer n such that $0.9^n < 10^{-200}$ is n = 4371.

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91影视

Find the smallest integer \(n\) such that \(0.9^{n}<10^{-200}\).

Short Answer

Step by step solution

Rewrite the inequality with the same base and apply the logarithm

Apply properties of logarithms and solve for n

Find the smallest integer n

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