The exponential functions make it possible to account for many phenomena with significant evolution. It is used to model population increases, to predict the decrease in radioactivity, the proliferation of such and such bacteria, the future behavior of financial markets, etc. In this article, we will discuss two ways of labeling an exponential function knowing at the start only the initial value of the phenomenon and its growth rate.
Method 1 of 2: Use the rate as a base
Step 1. Let's take an example
You deposit 1,000 euros in a bank account remunerated at 3% per year. Find an exponential equation that accounts for the evolution of this capital over time.
Step 2. Know the form of an exponential function
It can take the form: f (t) = P0(1 + r)t / h, where P0 is the initial value, t, the variable of time, r, the rate of change and h, the unit of time in connection with t.
Step 3. Make the digital application
Give to P0 and their respective values. The function then becomes: f (t) = 1 000 (1, 03)t / h.
Step 4. Find h
Remember your equation. You earn 3% interest every year, or if you prefer, every 12 months. If the placement time (t) is given in months, you divide t by 12. This is why, here, h = 12. The function then looks like this: f (t) = 1 000 (1, 03)t / 12. If the time units of the rate (r) and the placement time (t) are the same, h is equal to 1.
Method 2 of 2: Use "e" as the base
Step 1. Understand what “e” is
If you take "e" as the base, you are using what is called a "natural base", also called "Neper's constant". Its use makes it possible to directly deduce the continuous exponential growth from the equation.
Step 2. Let's take an example
You have a 500 g sample of a carbon isotope, which is known to have a half-life of 50 years, which is the time it takes for it to lose half of its mass.
Step 3. Know another form of the exponential function
It can also take the form: f (t) = aekt, where a is the initial value, e, the base, k, the continuous exponential growth, and t, the time variable.
Step 4. Replace the original value
This is the only constant value given to us from the start. Enter it into the equation, which gives: f (t) = 500thkt.
Step 5. Find the continuous exponential growth
This rate gives an indication of the speed of development of the function, both upwards and downwards. This speed is visible on the curve: the slope is more or less steep. If we take our example, we know that in 50 years, the sample loses half of its mass, i.e. 250 g. This date of 50 years can be considered as one of the points of the graph, which allows you to write: t = 50. Put this value in the function. We know that f (50) = 500th50k, but also that f (50) = 250. If we merge the two equations, we obtain the following exponential equation: 250 = 500th50k. It only remains to solve it. Divide on each side by 500, which gives: 1/2 = e50k. We take the natural logarithm (inverse function of the exponential) of the two members and we obtain: ln (1/2) = ln (e50k). It is a property of the logarithm which establishes that: ln (ex) = xln (e). Applied to our equation, this property allows us to write: ln (1/2) = 50k (ln (e)). We can write “ln” (natural log) or “log” (natural log). There is another property of logarithms which establishes that the log of the base is equal to 1, that is: ln (e) = 1. The equation then comes down to: ln (1/2) = 50k. Divide each side by 50, which gives: k = (ln (1/2)) / 50. With a calculator, you will find that: ln (1/2) = -0, 693147181. Divided by 50, we find that k is about -0, 01386. You will notice that this answer is negative. If the growth is negative, as here, it means that you have an exponential decrease (the sample loses its mass). If the growth is positive, you have an exponential increase.
Step 6. Replace k with its value
The function of the half-life (or half-life) is then the following: f (t) = 500e-0, 01386t.
- If you have a lot of calculations to do with your function, and you need great precision, you might want to save time by storing k in memory as well. You will enter its exact value with all the decimal places. It is a precaution to be taken if you want your machine to draw the curve of the function: do not use "x" to denote k. Indeed, if you did, you would end up with a bogus equation of function. Reserve, for example, "x" for t.
- With practice, you will quickly see which of the two methods is to be used. Most often, we take the first, but in some problems, it is better to use the second, the one with "e", the calculations are easier and faster.