For more details see the natural logarithm. For the specifics, ln is a special type of logarithm where the base is equal to the number, e. This substitution just makes the formula easier to read because ln(b) is a constant term. That is why I made the substitution C = ln(b). Really the only thing you need to know is that ln(b) will always be the same for any value of the functions main variable time, t. Firstly, let me explain the ln(b) term that appears in both expressions. You can see the two formula are drastically different. The derivative of the exponential formula is Īnd the derivative of the logarithmic formula is, Without getting side tracked into calculus, all you need to know is that the derivative of a formula merely describes that formula's rate of change from one time to the next. The primary difference between the two can been seen by looking at their respective derivatives. Where x(t) is the value of the formula as some time, t, the initial value is a, and b is a base of the logarithm and governs the rate of growth much like it does in the exponential function. If something is growing logarithmically than its growth is described by the formula, On the other hand we discussed that logarithmic growth is the exact opposite of exponential growth in the way that division or subtraction is the opposite of multiplication and addition. Where, a is the initial value, and b is the rate of growth. To quickly recap, in the last article we found that if something is growing exponentially, than the value of that "something", x(t), at some time in future, t, is given by the formula
The blue line is exponential growth and the purple logarithmic This graph shows the opposite nature of these two functions.
GUILD WARS 2 TIME SINK SERIES
In part one of the series I mainly answered the questions, what does it mean for something to grow exponentially? and what does it mean for something to grow logarithmically? Today I would like to tackle the final question I presented in the first part, why might wealth grow logarithmically in GW2?
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