Doubt in this question involving logarithm If $\log2= 0.301$, then how many number of digits are in $2^{64}$?
What I did:
$$\log(2)^{64}=\log2^{64}=64\log2=64\log2=19.264$$
Number of digits comes out to be $5$. But answer is $20$? I have written $\log 2$ raise to the power $64$

See that $100 = 10^2$ and $1000 = 10^3$ and so on.  The number of digits in both those numbers is one more than the exponent on $10$.  If we write any 3-digit number as a power of 10, then the exponent will be between $2$ and $3$.  For instance,  $423 = 10^{2.6263\ldots},$  where, of course $2.6263\ldots$ is the base-10 log of $423.$  
Using this principle we can do this:
$$2^{64}= \left(10^{\log 2}\right)^{64} = 10^{64\log } = 10^{19.26\ldots}. $$
So we know the number has 20 digits.
