Necessary Precision for x equal to log base 10 of 2 to determine Decimal expansion of 10 to the x

Theorem

Let $b = 10$.

Let $x \approx \log_{10} 2$.

Let it be necessary to calculate the decimal expansion of $x$ to determine the first $3$ decimal places of $b^x$.

An infinite number of decimal places of $x$ would in fact be necessary.

This is a trick question:

How many decimal places of accuracy of $x$ are needed to determine the first $3$ decimal places of $b^x$?

We have that $b^x = 10^{\log_{10} 2} = 2$.

Let $x_a < x < x_b$, where $x_a$ and $x_b$ are ever closer approximations to $x$.

Then:

$x_a$ begins $1 \cdotp 999 \ldots$

$x_b$ begins $2 \cdotp 000 \ldots$

and it will not be possible to achieve the full expansion of $b^x$.

$\blacksquare$