/*! 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 54 Suppose \(\log _{5}\left(\log _{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 54

Suppose \(\log _{5}\left(\log _{9} m\right)=6 .\) How many digits does \(m\) have?

Short Answer

Expert verified
The number \(m\) has 16,385 digits.

Step by step solution

01

Simplify the given equation

Rewrite the given equation as follows: \(\log _{5}\left(\log _{9} m\right)=6\)
02

Remove the outer logarithm

Let's remove the outer logarithm by applying exponentiation: \((\log _{9} m) = 5^6\)
03

Remove the inner logarithm

Now, remove the inner logarithm again by applying exponentiation: \(m = 9^{(5^6)}\)
04

Calculate the value of m

We can now calculate the value of \(m\): \(m = 9^{(5^6)} = 9^{15,\!625}\)
05

Determine the number of digits in m

To count the number of digits in \(m\), we can observe that: \(log _{10} m = log _{10} (9^{15,\!625})\) By the change of base formula: \(log _{10} m = \frac{log_{9} m}{log_{9} 10}\) Using the logarithmic properties: \(log _{10} m = \frac{15,\!625 \times log_{9} 9}{log_{9} 10}\) Since \(log_{9} 9 = 1\): \(log _{10} m = \frac{15,\!625}{log_{9} 10}\) As a result, the number of digits, \(n\), in \(m\) can be calculated as: \(n = \lfloor log _{10} m \rfloor + 1\) Plugging in the value from above: \(n = \lfloor \frac{15,\!625}{log_{9} 10} \rfloor + 1\) \(n = \left\lfloor \frac{15625}{0.954242} \right\rfloor + 1\) \(n \approx 16384 + 1 = 16385\) Therefore, the number \(m\) has 16,385 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

Find the number of digits in the given number. $$ 5^{999} \cdot 17^{2222} $$

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{4}(2 u v) $$

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

The 1995 earthquake in Kobe (Japan), which killed over 6000 people, had Richter magnitude 7.2. What would be the Richter magnitude of an earthquake that was 1000 times less intense than the Kobe earthquake?

How many more times intense is an earthquake with Richter magnitude 7 than an earthquake with Richter magnitude \(5 ?\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.