1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alla [95]
3 years ago
9

Find all n∈IN such that the number of digits of n equal to n

Mathematics
1 answer:
Butoxors [25]3 years ago
5 0
Note that \log_{10}1=0, \log_{10}10=1, \log_{10}100=2, and so on. The function \log_{10}x is continuous and increasing for all x>0, so when 1, we have 0; when 10, 1; and so on.

This means

\lfloor\log_{10}x\rfloor=\begin{cases}0&\text{for }1\le x

which means we can capture the number of digits of n\in\mathbb N with the function \lfloor\log_{10}n\rfloor+1.

So the problem is the same as finding positive integer solutions to

\lfloor\log_{10}n\rfloor+1=n

We know that n=1 has one digit, so clearly this must be a solution. We need to show that this is the only solution.

Recall that \dfrac{\mathrm d}{\mathrm dx}[\log_{10}x+1]=\dfrac1{\ln10\,x}, while \dfrac{\mathrm d}{\mathrm dx}[x]=1. This means \log_{10}x increases at a much slower rate than x as x\to\infty. We know the two functions intersect when x=1. Therefore it's clear that x>\log_{10}x+1 for all x>1.

Now, it's always the case that \lfloor f(x)\rfloor\le f(x), so we're essentially done:

x>\log_{10}x+1\ge\lfloor\log_{10}x\rfloor+1

which means there are no other solutions than n=1.
You might be interested in
There is a pattern (2,4) (4,6) (6,9) (8,12) what is the 8th group
Jobisdone [24]
The answer is (10,17)
3 0
3 years ago
Which might be a better way to evaluate 1150 divided by 46: drawing base-ten diagrams or using the partial quotients method ? Ex
hammer [34]

Answer:

idk sounds like a good question

Step-by-step explanation:

8 0
3 years ago
Y=x^2+3x+54 what is the table?
andrew11 [14]

Answer:

i added an attachment of the graph :)

Step-by-step explanation:

7 0
2 years ago
A boy had 3 apples and gained one. How many apples does he have now? ​
Zigmanuir [339]
Answer: 4



step by step explanation: 3+1=4
4 0
2 years ago
25000000 in standard form
Elan Coil [88]

Answer:

2,500,000 This is in standard form.

2.5 x 10^6  This is in scientific notation.

Step-by-step explanation:

6 0
3 years ago
Other questions:
  • parallel lines r and s are cut by two transversals, parallel lines t and u. Which angles are corresponding angles with angle 8?
    5·1 answer
  • Which expression represents the series 1+5+25+125+625?
    14·1 answer
  • What is the slope of the line that contains the cordon and points (8, -3) and (-2, 7)​
    11·1 answer
  • Is this graph a function or not a function?
    8·2 answers
  • Infinite Algebra 2
    15·1 answer
  • The lengths of two sides of triangle are given. Determine the range for the possible lengths of the third side
    6·1 answer
  • 1 and the
    6·1 answer
  • Help please it’s urgent I’ll mark brainliest
    7·2 answers
  • Brianna has 6 bags of soil. Filling one flowerpot requires 1/3
    5·2 answers
  • After 5 years, Ben’s income has undergone several increases. Last month, Ben put $600 in savings, which was 10% of his net incom
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!