Exponentials, logarithms

In the first section we explored the power A^B as a number and then we created the power function by fixing B and letting A change. Now we will try it the other way. We fix A and denote it a to emphasize it, and put x as a variable instead of B. What would be the domain of a function we thus create? To this end we have to reorganize the observations from the section on power as a number. There we looked at the powers from the point of view of B, but now the conclusions have to be looked at from the point of view of A, that is, a.

If a > 0, then for B we can take any real number. Thus we have a nice function a^x. If a = 0, we can again take any real number B, in this way we get the constant function 0^x = 1 (including the case 0⁰ = 1). However, this function is not included in the class we are investigating here since it is very different. For the same reason we do not include here the case a = 1, since 1^x = 1. If a < 0, then for B we can only take fractions whose denominators are odd. However, the set of such numbers (although dense on the real line) is very scattered and does not contain any interval. Functions with such domains are pretty much useless from the point of view of analysis, since most methods require functions defined on non-degenerate intervals and their unions. Thus we also do not include a < 0.

In conclusion, we define the general exponential only for a > 0, a ≠ 1.

Definition.
By a general exponential function with base a > 0, a ≠ 1, we mean the function a^x.
By the exponential function we mean the function e^x, where e is the Euler number 2.718281828...

Sometimes we also write exp(x) or Exp(x) instead of e^x, especially if the argument is a more complicated expression.

The number e has to be defined somehow. The most common definition is using a limit, the Euler number is defined as the limit of the sequence (1 + 1/n)ⁿ, see Sequences - Theory - Important examples.

The general exponential function has real numbers as the domain. The algebraic properties of powers translate to some useful properties of the general exponential function, for all real x,y, and c we have

a^x + y = a^x⋅a^y;       a^c⋅x = (a^x)^c = (a^c)^x

The general exponential is monotone, namely increasing for a > 1 and decreasing for a between 0 and 1. In both cases the general exponential is concave up. There is a natural ordering between the general exponential functions. The inequality a < b implies a^x < b^x for x positive and b^x > a^x for x negative.

The graphs are like this:

We see that the range is R(a^x) = (0,∞). If a > 1, then

For a∈(0,1) we have

For a > 0, a ≠ 1, the general exponential a^x is a monotone function on its domain and therefore 1-1. Thus it has an inverse. This inverse is called the logarithm with base a and denoted log_a(x). That is, for a positive x, the log_a(x) is such a number y that a ^y = x.

The special case, when a = e, is called the natural logarithm and is denoted ln(x). In engineering, the logarithm with base a = 10 is often used, it is called the decadic logarithm and usually denoted log(x).

The properties of logarithm naturally follow from properties of the general exponential. There are two important algebraic identities:

log_a(x⋅y) = log_a(x) + log_a(y);       log_a(x^c) = c⋅log_a(x).

By the way, although many students get very creative during tests, there is really no identity for log_a(x + y).

The domain of a logarithm is (0,∞), the range is the whole real line and log_a(1) = 0. Logarithm is monotone -- increasing for a > 1, decreasing for a∈(0,1) -- and concave down. Important limits at endpoints are for a > 1:

and for a∈(0,1):

This follows easily from the graphs:

Again, there is a natural ordering, but this time it is more complicated and depends on comparing the base to 1. If a < b < 1 or 1 < a < b, then log_a(x) < log_b(x) for x∈(0,1) and log_b(x) < log_a(x) for x > 1. If a < 1 < b, then log_b(x) < log_a(x) for x∈(0,1) and log_a(x) < log_b(x) for x > 1.

The fact that a logarithm is the inverse to the appropriate general exponential can be expressed by these important equalities:

x = log_a(a^x) for x real     and     x = a^log_a(x) for x > 0.

When applied to the natural logarithm, these equalities say that x = ln(e^x) for real x and x = e^ln(x) for positive x. The latter equality is very useful, here in Math Tutor we call it the "e to ln trick". In particuar, using it one can deduce the following identities which show that it is in fact enough to know the exponential (with the base e) and the natural logarithm, since for every a > 0 we have

Indeed, the second equality requires just a little thought and the first equality is even easier, we will show it two ways:

We conclude this section with formulas for derivatives:

Interesting remark: There are several ways in which to define the exponential. Mathematicians like abstract things and so they often use this version:

Exponential is the function f defined on a real line that satisfies f (0) = 1, f ′(0) = 1, and f (x + y) = f (x)⋅f (y) for all x,y.

Of course, one has to prove that there is only one such function, but this can be done, and since our exponential has these properties, it must be the one. Similarly one can define the general exponential by changing the second condition to f ′(0)=ln(a).

General powers
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