Converting base2 scientific notation to base10 scientific notation I would now like to know how to convert a value in base2 scientific notation (is that the correct terminology?), say 1.93 * 2 ^ 88, into the form of A * 10 ^ B.
I want to do this without expressing the first form as a complete number and converting it back to the 2nd form.
I can get by exponent by doing 88 * (ln2 / ln10) = 26.49...
So I have A * 10 ^ 26 now.
How can I solve for A?
Complete equation: 1.93 * 2 ^ 88 = A * 10 ^ 26.

This has been moved/duplicated from https://softwareengineering.stackexchange.com/questions/231692/converting-base2-scientific-notation-to-base10-scientific-notation.

I would take a slightly different approach: solve for both the significand and exponent concurrently.
Suppose the original number is $S \times 2^n.$
First convert the entire number into a base-ten logarithm like this:
\begin{align}
X &= \log_{10} (S \times 2^n) \\
  &= \log_{10} (S) + \log_{10}(2^n) \\
  &= \log_{10} (S) + n\log_{10}(2)  \\
  &= \frac{\ln(S)}{\ln(10)} + n \frac{\ln(2)}{\ln(10)}.
\end{align}
Now the original number is $10^X.$ To express this in scientific notation, let
\begin{align}
k &= \lfloor X \rfloor, \\
M &= X - k, \\
A &= 10^M,
\end{align}
and in this way we find that $S \times 2^n = A \times 10^k.$
Applying this to your example,
\begin{align}
X &= \log_{10}(1.93 \times 2^{88})  \\
  &= \frac{\ln(1.93)}{\ln(10)} + 88 \frac{\ln(2)}{\ln(10)} \\
  &\approx 0.285557 + 88 \times 0.301030 \\
  &\approx 26.7762
\end{align}
Since $10^{0.7762} \approx 5.97,$ the result is
$$ 1.93 \times 2^{88} = 5.97 \times 10^{26}. $$
