/*! 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 21 Suppose \(\log _{8}\left(\log _{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 21

Suppose \(\log _{8}\left(\log _{7} m\right)=5 .\) How many digits does \(m\) have?

Short Answer

Expert verified
The integer \(m\) has 27694 digits.

Step by step solution

01

Transform the equation to exponential form using base 8

We are given the equation \(\log _{8}\left(\log _{7} m\right)=5\). The first step is to rewrite this equation as an exponential form using the base \(8\): \[8^5 = \log _{7} m\]
02

Find the value of m

Now, we need to rewrite this equation back into logarithmic form using the base \(7\) to determine the value of \(m\). Before that, let's calculate the value of \(8^5\): \[8^5 = 32768\] So the equation becomes: \[\log _{7} m = 32768\] Now we can rewrite this into the exponential form with base \(7\): \[m = 7^{32768}\]
03

Determine the number of digits in m

To find the number of digits in \(m = 7^{32768}\), we can use logarithm properties. Let's find the closest integer \(n\) such that \(10^n\) is less than or equal to \(m\), i.e., \(10^n \leq 7^{32768}\). Take logarithm base 10 on both sides: \[\log_{10} (10^n) \leq \log_{10} (7^{32768})\] Using logarithm property \(\log_a (b^c) = c\log_a b\), we get: \[n \leq 32768 \log_{10}7\] Now, we can approximate the value of \(\log_{10}7 \approx 0.845\), so we get: \[n \leq 32768 * 0.845\] \[n \leq 27694.75\] Thus, the number of digits in \(m\) is the largest integer not greater than \(27694.75\), which means that \(m\) has 27694 digits.

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

For \(x=18\) and \(y=0.3,\) evaluate each of the following: (a) \(\ln \frac{x}{y}\) (b) \(\frac{\ln x}{\ln y}\)

Estimate the given number. Your calculator will be unable to evaluate directly the expressions in these exercises. Thus you will need to do more than button pushing for these exercises. \(\left(1-\frac{2}{8^{99}}\right)^{\left(8^{99}\right)}\)

Show that for every positive number \(c,\) we have $$ \ln (c+t)-\ln c \approx \frac{t}{c} $$ for small values of \(t\).

Find all numbers \(x\) that satisfy the given equation. \(e^{x}+e^{-x}=6\)

(a) Using a calculator, verify that $$ \log (1+t) \approx 0.434294 t $$ for some small numbers \(t\) (for example, try \(t=0.001\) and then smaller values of \(t\) ). (b) \(\quad\) Explain why the approximation above follows from the approximation \(\ln (1+t) \approx t\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure 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.