Smallest power of 2 consisting of N digits

Given an integer N, the task is to find the smallest power of 2 which consists of N digits.

Examples:

Input: N = 3
Output: 7
Explanation:
2⁷ = 128, which has three digits.

Input: N = 4
Output: 10
Explanation:
2¹⁰ = 1024, which has four digits.

Naive Approach: A simple solution is to iterate through all the powers of 2 starting from 2⁰ and check for each power of 2, if it contains N digits or not. Print the first power of two which contains N digits.

Below is the implementation of the above approach:

C++

// C++ Program to implement
// the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to return smallest
// power of 2 with N digits
int smallestNum(int n)
{
    int res = 1;

    // Iterate through all
    // powers of 2
    for (int i = 2;; i *= 2) {
        int length = log10(i) + 1;
        if (length == n)
            return log(i) / log(2);
    }
}

// Driver Code
int main()
{
    int n = 4;
    cout << smallestNum(n);

    return 0;
}

Java

// Java program to implement
// the above approach
import java.io.*;

class GFG{

// Function to return smallest
// power of 2 with N digits
static int smallestNum(int n)
{
    int res = 1;

    // Iterate through all
    // powers of 2
    for(int i = 2;; i *= 2)
    {
        int length = (int)(Math.log10(i)) + 1;

        if (length == n)
            return (int)(Math.log(i) /
                         Math.log(2));
    }
}

// Driver Code
public static void main (String[] args)
{
    int n = 4;

    System.out.print(smallestNum(n));
}
}

// This code is contributed by code_hunt

Python3

# Python3 program to implement
# the above approach
from math import log10, log

# Function to return smallest
# power of 2 with N digits
def smallestNum(n):

    res = 1

    # Iterate through all
    # powers of 2
    i = 2
    while (True):
        length = int(log10(i) + 1)

        if (length == n):
            return int(log(i) // log(2))

        i *= 2

# Driver Code
n = 4

print(smallestNum(n))

# This code is contributed by SHIVAMSINGH67

C#

// C# program to implement
// the above approach
using System;

class GFG{

// Function to return smallest
// power of 2 with N digits
static int smallestNum(int n)
{
    //int res = 1;

    // Iterate through all
    // powers of 2
    for(int i = 2;; i *= 2)
    {
        int length = (int)(Math.Log10(i)) + 1;

        if (length == n)
            return (int)(Math.Log(i) /
                         Math.Log(2));
    }
}

// Driver Code
public static void Main ()
{
    int n = 4;

    Console.Write(smallestNum(n));
}
}

// This code is contributed by code_hunt

Javascript

<script>

// Javascript Program to implement
// the above approach

// Function to return smallest
// power of 2 with N digits
function smallestNum(n)
{
    res = 1;

    // Iterate through all
    // powers of 2
    for (var i = 2;; i *= 2) {
        var length = parseInt(Math.log(i)/Math.log(10)) + 1;
        if (length == n)
            return parseInt(Math.log(i) / Math.log(2));
    }
}

// Driver Code
n = 4;
document.write(smallestNum(n));

</script>

Output:

10

Time Complexity: O(log₂n)
Auxiliary Space: O(1)

Efficient Approach: The key observation to optimize the above approach is that the smallest power of 2 with N digits can be obtained by the equation:

Below is the implementation of the above approach:

C++

// C++ Program of the
// above approach

#include <bits/stdc++.h>
using namespace std;

// Function to return smallest
// power of 2 consisting of N digits
int smallestNum(int n)
{
    float power = log2(10);
  cout<<power;
    return ceil((n - 1) * power);
}

// Driver Code
int main()
{
    int n = 4;
    cout << smallestNum(n);

    return 0;
}

Java

// Java Program of the
// above approach
class GFG{

  // Function to return smallest
  // power of 2 consisting of N digits
  static int smallestNum(int n)
  {
    double power = log2(10);
    return (int) Math.ceil((n - 1) * power);
  }

  static double log2(int N)
  {
    // calculate log2 N indirectly
    // using log() method
    return (Math.log(N) / Math.log(2));
}

// Driver Code
public static void main(String[] args)
{
  int n = 4;
  System.out.print(smallestNum(n));

}
}

// This code is contributed by Princi Singh

Python3

# Python3 program of the
# above approach
from math import log2, ceil

# Function to return smallest
# power of 2 with N digits
def smallestNum(n):

    power = log2(10)
    print(power);
    return ceil((n - 1) * power)

# Driver Code
n = 4

print(smallestNum(n))

# This code is contributed by SHIVAMSINGH67

C#

// C# program of the
// above approach
using System;
class GFG{

// Function to return smallest
// power of 2 consisting of N digits
static int smallestNum(int n)
{
  double power = log2(10);
  return (int) Math.Ceiling((n - 1) * power);
}

static double log2(int N)
{
  // calculate log2 N indirectly
  // using log() method
  return (Math.Log(N) / Math.Log(2));
}

// Driver Code
public static void Main()
{
  int n = 4;
  Console.Write(smallestNum(n));
}
}

// This code is contributed by Chitranayal

Javascript

<script>

// JavaScript program for
// the above approach

 // Function to return smallest
  // power of 2 consisting of N digits
  function smallestNum(n)
  {
    let power = log2(10);
    return Math.ceil((n - 1) * power);
  }

  function log2(N)
  {

    // calculate log2 N indirectly
    // using log() method
    return (Math.log(N) / Math.log(2));
    }

// Driver code
    let n = 4;
   document.write(smallestNum(n));

   // This code is contributed by splevel62.
</script>

Output:

10

Time Complexity: O(1) 
Auxiliary Space: O(1)