Rewriting Googol (10¹⁰⁰) as a Power with Base 2

If you want to figure out how many bits it would take to represent Googol as a binary integer, you need to first rewrite Googol as a power with base 2. The exponent (rounded up) represents the number of bits it will take to represent that number. For example, the number of bits needed to represent 10 as a binary integer is 4. We know this because 2^~3.3 and so 3 rounded up is 4. 10_d = 1010_b.

So let's write 10¹⁰⁰ as a power with base 2:

10¹⁰⁰ = 2^x

Then solve for x. Take the log of both sides and then isolate:

  log₁₀(10¹⁰⁰) = log₁₀(2^x)
100log₁₀(10) = xlog₁₀(2)
           100(1) = xlog₁₀(2)
                     x = ¹⁰⁰/_log10(2)

So then we can say that:

10¹⁰⁰ = 2^{100/_log10(2)}
10¹⁰⁰ = 2^~332.2

So 333 bits are needed to represent Googol as a binary integer.