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To find how long it takes for 9000 bacteria, we replace the B with 9000 and solve for t:
To solve for t we'll start by dividing both sides by 100 (to get the equation into exponential form):
Next we use logarithms (for reasons that will become clear later). While any base logarithm may be used, the solution will be simpler if we use base e (aka ln) logarithms:
On the right side we can use a property of logarithms,
, to move the exponent out in front of the logarithm. (It is this ability to "move" an exponent like this that is the reason we use logarithms. Moving the exponent out in front puts the variable in a place "where we can get at it".)
By definition, ln(e) = 1 (This is why we chose base e logarithms.), so this simplifies to:
Now we can multiply both sides by 2 (the reciprocal of 0.5 = 1/2):
This is an exact expression for the answer. For a decimal approximation, get out your calculator to find ln(90). (And then multiply that by 2!).
If your calculator does not have an "ln" button, then the base conversion formula,
to convert the base e log into base 10 logs:
And, if your calculator does not have ln then it probably does not have a button for "e" either. For "e" use 2.7182818284590451 (or a rounded off version of it):