$ 10^{-9}[2\times10^6 + 3^{1000}] $

Question

$$ 10^{-9}[2\times10^6 + 3^{1000}] $$

I'm stuck on solving this. I wasn't able to put this into my calculator since the number is too big for it to calculate.

So far I've done this:

$$ 2\times10^{-3} + 3^{1000}\times10^{-9} $$ log the equation $$ \log[2\times10^{-3} + 3^{1000}\times10^{-9}] $$ $$ \log(2\times10^{-3}) + \log(3^{1000}\times10^{-9}) $$ $$ (\log\,2 + log\,10^{-3}) + (\log\,3^{1000} + \log \, 10^{-9}) $$ $$ (\log \, 2 - 3) + (1000 \times \log \, 3 - 9) $$

How do I get rid of the log afterwards?

Answer 1

distribute the sum as $ 10^{-9}[2\times10^6] + 10^{-9}\times 3^{1000} $, and write $3^{1000}$ as $10^{{1000}\log_{10}{3}}$

Answer 2

You have to decide what form you want the answer in. You can type it into Alpha and get a number with $469$ digits, but that is not very useful. One of the big points of the exercise was to see that the $2 \times 10^6$ is completely negligible compared to $3^{1000}$, so ignore it. In fact, it is smaller by a factor of more than $10^{470}$. Then you can write it as $10^{1000 \log_{10}3-9}\approx 10^{468}$ and I would leave it like that.