Get the number of digits of a power of 2, with large exponents?

Question

I am doing the following exercise: The number of digits of a power of 2.. The statement is:

What is the number of digits of a power of 2?

2 ---> 1 digit 
2 * 2 = 4 ---> 1 digit 
2 * 2 * 2 = 8 ---> 1 digit
  2 * 2 * 2 * 2 = 16 ---> 2 digits
  ... ... ... 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024 ---> 4 digits

Then, given the exponent, what would be the number of digits of that power?

I have tried the following answer:

import java.math.BigInteger; 
public class Power {
    public static long digit(long exp) {
    System.out.println("exp: "+exp);
    BigInteger pow = BigInteger.valueOf(2).pow((int)exp);
    return String.valueOf(pow).split("").length;
    }
}  

However it times out with big exponents like: 562078812

How could we improve this solution? Is there any fastest answer?

I have also read:

• https://www.geeksforgeeks.org/biginteger-pow-method-in-java/
• BigInteger.pow(BigInteger)?
• Infinite Loop During Calculation of Power of Big Integers Java

Answer 1

The fastest answer is to use math. The number of digits in 2^n is (nlog₁₀2)+1 . You can achieve that by simply returning n * Math.log10(2) + 1. Good luck.