The number e has been called one of the most important numbers in all of mathematics. However, it is important to remember that e is just a number. Calculated to nine decimal places,
e = 2.718281828
e can be extended to countless decimal places and no patterns have ever been discovered. In this sense e, is very similar to pi.
Look at these two graphs. The first is the graph of y = e^x and the second is y= e^-x
Notice in the first graph, to the left of the y-axis, e^x increase very slowly, it crosses the axis at y = 1, and to the right of the axis, it grows at a faster and faster rate.
The second graph is just the opposite. For negative x's, the graph decays in smaller and smaller amounts. It crosses the y-axis at y = 1, and then decays at slower and slower rates.
The natural log is the logarithm whose base is e. The two functions, the natural log and the exponential e, are inverses of each other. In other words, saying y = Ln[x] is the same as e^y = x.
Look at the plot of y = Ln[x].
The logarithm grows fast at first, then gradually slows. It also crosses the x-axis at 1 and can only be found for x > 0. Therefore, Log = 0, Ln[0 < x < 1] = - number, and the Ln[x < 0] does not exist. In other words, you cannot take the natural log of a negative number.
Here are some important properties of exponential and log functions, you may find useful.
e^(a + b) = e^a * e^b
e^(a - b) = e^a / e^b
(e^a)^b = e^(a * b)
Log[a * b] = Log[a] + Log[b]
Log[a / b] = Log[a] - Log[b]
Log[a^b] = b Log[a]
The exponential function can be described as,
where a and b are constants. The curve that we use to fit data sets is in this form so it is important to understand what happens when a and b are changed.
Recall that any number or variable when raised to the 0 power is 1. In this case if b or x is 0 then, e^0 = 1. So at the y-intercept or x = 0, the function becomes y = a * 1 or y = a. Therefore, the constant a is the y-intercept of the curve.
The other parameter in our equation is b. If b is very small and greater than 0, the function flattens out. The curve increases at a slower rate then for large b's. On the contrary, for large b's the curve increases quickly.
Look at these two plots. The first is for an equation with a large b, and the second is for a small b. Notice the scales of the plots.
For, b's less than 0, the same occurs except the plots look like the plot of e^-x from above.
1.)Simplify the following expressions.
a.)e^(ln 2 + ln x) = 2x
b.)ln(e^(1/x)) = 1/x
2.)Solve for y.
a.)e^(2y) = x^2 = ln(x)
b.)ln(y - 1) = x + ln x = xex + 1
3.)Sketch the following curves on the same axes. Identify the domains of each equation in terms of x.
a.)y = ln(-x) and y = -ln(x).
b.)y = e^(-x) and y = -e^(x).
Application of Exponentials
4.)If you invest A dollars at a fixed annual interest rate, r and interest is compounded continuously to your account, the amount of money, Ao, you will have at the end of t years is,
Compounded continuously means that the money in your account is continuously being added interest. It can almost be said that the interest is being added every second, day or night.
a.)You deposit $621 in an account that pays 10% interest. How much money will you have after 8 years? after 10 years?
8 years: 621e0.8 = $1382.061
10 years: 621e = $1688.053
b.)How long will it take you to double your money if you invest $500 at an interest rate 6%? = (0.06)/ln(2) = 11.552 years
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