/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Find the smallest integer \(n\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write the series using summation notation (starting with \(k=1\) ). Each series is either an arithmetic series or a geometric series. $$ 2+4+6+\cdots+100 $$

Express $$ 8.237545454 \ldots $$ as a fraction; here the digits 54 repeat forever.

Show that $$ \ln n<1+\frac{1}{2}+\cdots+\frac{1}{n-1} $$ for every integer \(n \geq 2\). [Hint: Draw the graph of the curve \(y=\frac{1}{x}\) in the \(x y\) plane. Think of \(\ln n\) as the area under part of this curve. Draw appropriate rectangles above the curve.]

Explain why the polynomial factorization $$ 1-x^{n}=(1-x)\left(1+x+x^{2}+\cdots+x^{n-1}\right) $$ holds for every integer \(n \geq 2\).

Write the series using summation notation (starting with \(k=1\) ). Each series is either an arithmetic series or a geometric series. $$ \frac{5}{9}+\frac{5}{27}+\frac{5}{81}+\cdots+\frac{5}{3^{40}} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.