From an expression raised in a power of 2 to an expression raised in the power or 10

Question

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent.

Example:
I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from the result I "extracted" the $~1.7\cdot 10^38$

Is there a way for these cases to avoid doing the actual multiplication?

Answer 1

Take $\log _2\left( x \right)$, then multiply by $\log _{10}\left( 2 \right)$ and the answer is 10^whatever you got.

So $10^{\log _2\left( x \right)\log _{10}\left( 2 \right)}$

Reason: $\log _b \left( x \right) = \log _b \left( c \right)\log _c \left( x \right) = \frac{{\log _c \left( x \right)}}{{\log _c \left( b \right)}}$