Indices Problem?

Question

The problem/question

A microbiologist isolates a single bacterium in a petri dish and observes that it divides itself into two every hour.

How long would it take for the backteria culture to contain approximately 1.15x10^18 bacteria?

I really appreciate if you could show me your working out.

Thanks {{^_^}}

Answer 1

You start off with one Bacterium.
0 hour .......= 1 Bacterium.
1hour later = 2 Bacteria
2 hours later = 4 Bacteria
3 hours later = 8 Bacteria. (and etc.).

Now look at the number pattern.
2° = 1
2` = 2
2² = 4 and etc.

2ⁿ = 1∙15 x 10^18 (takes logs).
Log 2ⁿ = Log (1∙15 x 10^18) (relocate the power).
n Log 2 = Log (1∙15 x 10^18)
n = Log (1∙15 x 10^18) / (Log 2)
n = 18∙060 697 84... / 0∙301 029 995...
n = 59∙996 339 57

Time is : 59∙996..
Time is ≈ 60 hours.

Answer 2

This is a exponential growth problem. A rough approximation is based on the "fact" that 2^40 is appx 1.1 x 10^12. and 2^10 is appx 1000. On that basis, you wind up with about 61 hours (1 hour to go from 2^0 to 2^1).

Answer 3

You are trying to solve the problem of

2^x = 1.15x10^18

log2^x = log1.15x10^18

xlog2 = log1.15x10^18

x = log1.15x10^18 / log 2

x = 59.996 hours

So 60 hours would be about it.