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Stuck on logarithms question

Logarithms 2.png

I'm stuck on art b and c.

For b, I've tried making the formula equal to 2000.

For c, I don't understand how the powers are negative.
Original post by Ameba
For b, I've tried making the formula equal to 2000.


p is measued in *thousands* so a population of 2000 implies that p=2 instead.
So you can proceed to try and solve the formula being equal to 2 for t. At some point you would be required to take the logarithm of a negative number, which is not possible, which implies there is no solution, which implies there is never a time when the population is 2000.

EDIT: In fact in this case you won't be needing to take log of a -ve because at the preceding step you would be solving a quadratic that has no real solutions. This has the same implications; population is never 2000.


For c, I don't understand how the powers are negative.


Compare the form they give you in part (c) to the form given initially. How is the denominator different?? What can you divide the denominator in the initial form by in order to obtain the denominator in the form from part (c) ?? Do the same to the numerator and it should simplify to the numerator given in part (c)
(edited 4 years ago)
Not quite a good enough argument, since the numerator increases as well as the denominator. And we also aren't really interested in long-term behaviour. It definitely goes to zero, but it *could* be the case that the population grows and reaches 2000 at some point before decaying down to 0 in the long term.

If the numerator was a constant or decreasing function, then I'd say your justification is better suited.
(edited 4 years ago)
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Original post by RDKGames

Compare the form they give you in part (c) to the form given initially. How is the denominator different?? What can you divide the denominator in the initial form by in order to obtain the denominator in the form from part (c) ?? Do the same to the numerator and it should simplify to the numerator given in part (c)


That's what I'm confused on. I don't get how they got the denominator.
Original post by Ameba
That's what I'm confused on. I don't get how they got the denominator.


Well e−0.3te^{-0.3t} is the same as 1e0.3t\dfrac{1}{e^{0.3t}}

So the denominator goes from e0.3t+1e^{0.3t}+1
to 1+1e0.3t1+\dfrac{1}{e^{0.3t}}.

Hopefully it's obvious now what has been done...

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