MHB How Many Digits Are in $19^{9^9}$?

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    2017
AI Thread Summary
To determine the number of digits in the integer $19^{9^9}$, the formula for the number of digits in a number $n$ is given by $\lfloor \log_{10} n \rfloor + 1$. Applying this to $19^{9^9}$ involves calculating $\log_{10}(19^{9^9})$, which simplifies to $9^9 \cdot \log_{10}(19)$. The value of $9^9$ is a large number, and multiplying it by $\log_{10}(19)$ provides the total logarithmic value needed to find the digit count. The final result will yield the exact number of digits in $19^{9^9}$.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

How many digits are there in (decimal representation of) the integer $19^{9^9}$?

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
No one answered last week's problem.(Sadface)

You can find the suggested solution below:
Note that $19^{9^9}=10^{9^9\log 19}=10^{9^9 \cdot 1.27875} \approx 10^{495,415,345.4}$, therefore the number $19^{9^9}$ requires $495,415,346$ digits.
 

Similar threads

MHB Perfect Squares as Divisors of $1!\cdot 2! \cdot 3! \cdots 9!$
Replies
1
Views
1K
MHB What Are the Last Two Digits of the Sum of Factorials from 7! to 2016!?
Replies
1
Views
2K
MHB Integral Sum Problem #272: Find the Integral Part of a Sum
Replies
1
Views
1K
MHB What Is the Smallest k for a Cubic Function to Have Exactly One Real Root?
Replies
1
Views
2K
MHB Integers as Roots of Polynomial: Find Possible Values | POTW #293 Dec 19th, 2017
Replies
2
Views
1K
MHB What Is the Missing Digit in $2^{29}$?
Replies
1
Views
1K
MHB Minimum Value of $(u-v)^2$ and $\left(\sqrt{2-u^2}-\dfrac{9}{v}\right)^2$
Replies
1
Views
1K
MHB What Is the Sum of the Digits of 3n Given These Conditions?
Replies
1
Views
1K
MHB What Is the Smallest Integer $a$ for $\frac{1}{640}=\frac{a}{10^b}$?
Replies
1
Views
1K
MHB Can You Prove $ab\le 1$ Given $(x+a)(x+b)=9$?
Replies
1
Views
1K

Recent Insights

Back
Top