Ralzereep9h

2022-10-24

Finding out the logarithmic function for the situation below
There are 3000 barbs in a pond and every year 20% of the barbs die and then 1000 new barbs come to the pond. A logarithmic function needs to be plotted to graph this change in population.
I worked through a part of the above situation and arrived at the function:
$y=3000\left({0.8}^{x}\right)+1000\left({0.8}^{x-1}\right)+{0.8}^{x-2}+\cdots +{0.8}^{1}+{0.8}^{0}$
How do I convert this above equation into a logarithmic function?

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indivisast7

Expert

If you work out the numbers for about 30 years you will see that the values get closer and closer to 5000 from below without reaching it. Another way to find the special value 5000 is to ask which value of y will cause no change in the population the next year. This gives the equation $y=0.8y+1000$ which has the solution $y=5000$
So if you consider the "base line" to be $5000$ you can reword the expression as:
Each year $20$% of the difference between the current value and $5000$ is added.
$y=5000-2000\cdot {0.8}^{x}$
Note, however, that this is an exponential expression, not a logarithmic one. You do get a logarithm if you solve for $x$, namely
$x={\mathrm{log}}_{0.8}\frac{5000-y}{2000}$
or perhaps
$x=\frac{\mathrm{ln}\left(5000-y\right)-\mathrm{ln}2000}{\mathrm{ln}0.8}$

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