Given A Number n in a base 10, find the number of digits in its base b representation.
Constraints : 
Whole
Examples :
Input : Number = 48
Base = 4
Output: 3
Explanation : (48)10 = (300)4
Input : Number = 1446
Base = 7
Output: 4
Explanation : (446)10 = (4134)7
A simple approach: convert the decimal number into the given base r and then count number of digits.
An efficient approach : It resides on the relationship between the base of the number and number of digits of that number.
Typically : Let n be a positive integer. The base 
representation of 
has 
digits if 
, which is the case if 
or 
.The number of digits in the base b representation of n is therefore
In above equation the base changing logarithmic property has been used. So we calculate the logarithm of the number in that base which we want to calculate the number of digits. And take its floor value and then add 1.
This idea can be further used to find the number of digits of a given number n of base b in base r. All have to be done is to convert the number in base 10 and then apply the above formula of finding digits. It would be easier to calculate log of any base when number is in base 10.
C++
// C++ program to Find Number of digits // in base b. #include <iostream> #include <math.h> using namespace std; // function to print number of // digits void findNumberOfDigits(long n, int base) { // Calculating log using base // changing property and then // taking it floor and then // adding 1. int dig = (int)(floor( log(n) / log(base)) + 1); // printing output cout << "The Number of digits of " << "Number " << n << " in base " << base << " is " << dig; } // Driver method int main() { // taking inputs long n = 1446; int base = 7; // calling the method findNumberOfDigits(n, base); return 0; } // This code is contributed by Manish Shaw // (manishshaw1) |
Java
// Java program to Find Number // of digits in base b. class GFG { // function to print number of digits static void findNumberOfDigits(long n, int base) { // Calculating log using base changing // property and then taking it // floor and then adding 1. int dig = (int)(Math.floor( Math.log(n) / Math.log(base)) + 1); // printing output System.out.println("The Number of digits of Number " + n + " in base " + base + " is " + dig); } // Driver method public static void main(String[] args) { // taking inputs long n = 1446; int base = 7; // calling the method findNumberOfDigits(n, base); } } |
Python3
# Python3 program to Find Number of digits # in base b. import math # function to print number of # digits def findNumberOfDigits(n, base): # Calculating log using base # changing property and then # taking it floor and then # adding 1. dig = (math.floor(math.log(n) / math.log(base)) + 1) # printing output print ("The Number of digits of" " Number {} in base {} is {}" . format(n, base, dig)) # Driver method # taking inputs n = 1446base = 7 # calling the method findNumberOfDigits(n, base) # This code is contributed by # Manish Shaw (manishshaw1) |
C#
// C# program to Find Number of digits // in base b. using System; class GFG { // function to print number of // digits static void findNumberOfDigits(long n, int b) { // Calculating log using base // changing property and then // taking it floor and then // adding 1. int dig = (int)(Math.Floor( Math.Log(n) / Math.Log(b)) + 1); // printing output Console.Write("The Number of digits" + " of Number " + n + " in base " + b + " is " + dig); } // Driver method public static void Main() { // taking inputs long n = 1446; int b = 7; // calling the method findNumberOfDigits(n, b); } } // This code is contributed by Manish Shaw // (manishshaw1) |
PHP
<?php // PHP program to Find Number // of digits in base b. // function to print // number of digits function findNumberOfDigits($n, $b) { // Calculating log using base // changing property and then // taking it floor and then // adding 1. $dig = (int)(floor(log($n) / log($b)) + 1); // printing output echo ("The Number of digits". " of Number ". $n. " in base ".$b. " is ".$dig); } // Driver Code $n = 1446; $b = 7; // calling the method findNumberOfDigits($n, $b); // This code is contributed by // Manish Shaw (manishshaw1) ?> |
Output :
The Number of digits of Number 1446 in base 7 is 4
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