Finding out the logarithmic function for the situation below
The situation reads as follows:
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 ( 0.8 x ) + 1000 ( 0.8 x 1 ) + 0.8 x 2 + + 0.8 1 + 0.8 0
How do I convert this above equation into a logarithmic function?

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Answer & Explanation




2022-10-25Added 13 answers

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 = 0.8 y + 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.
This leads to the formula
y = 5000 2000 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 = log 0.8 5000 y 2000
or perhaps
x = ln ( 5000 y ) ln 2000 ln 0.8

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