Number of Digits in Power of 2/Examples/Mersenne Number M127

Example of Number of Digits in Power of 2
When expressed in conventional decimal notation, the number of digits in the Mersenne number $M_{127}$ is $39$.

Proof
Let $m$ be the number of digits in the Mersenne number $M_{127}$.

Recall the definition Mersenne number $M_{127}$:
 * $M_{127} = 2^{127} - 1$

We have that $2^{127}$ is not a power of $10$.

Neither can $2^{127} - 1$ be a power of $10$.

So $M_{127}$ and $2^{127}$ have the same number of digits.

From Number of Digits in Power of 2:
 * $m = \ceiling {127 \log_{10} 2}$

From Common Logarithm of 2:
 * $\log_{10} 2 \approx 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$

and so:
 * $m = \ceiling {38 \cdotp 23}$

Hence the result.