# Exponential and logarithmic functions

The next couple of weeks will be about exponential and logarithmic functions. Logarithms are particularly useful in many applications of mathematics.

## Exponential functions

I'll introduce exponential functions using a couple of analogies with other kinds of functions.

A power function is a function f of the form

• f(x) = xn,
for some constant n called the exponent of the function; we've seen lots of examples of power functions up to now. In contrast, an exponential function is a function f of the form
• f(x) = bx,
for some constant b called the base of the function. The base should be a positive number, so that bx makes sense for every real number x.

Now recall that a linear function is a function f of the form

• f(x) = mx + b,
for some constants m (the rate of change) and b (the initial value). Analogously, a generalized exponential function is a function f of the form
• f(x) = Cbx,
for some constants b (the base) and C (the initial value).

If you don't remember any other values of a generalized exponential function, remember these:

• f(0) = Cb0 = C,
• f(1) = Cb1 = C · b,
• f(−1) = Cb−1 = C/b.
The domain of a generalized exponential function is the set of all real numbers; as long as b ≠ 1, the range of the exponential function with that base is the set of all positive numbers. (Because b is positive, bx is also positive.) Then the range of a generalized exponential function is the set of all real numbers with the same sign as C. (If b = 1 or C = 0, then the range consists of only C.) If C > 0 and b > 1, then the generalized exponential function is increasing; if either of these is reversed, then the function is decreasing; if both are reversed, then it's increasing again. (If b = 1 or C = 0, then the function is constant.)

Besides numbers such as 10, 2, and 1/2, which you are familiar with, another common choice of base is a special number, about 2.72, known as e. The importance of this number e will become clear when we look at applications. Many calculators have a button that calculates ex from x; in particular, you can calculate e itself, as e1, using this button.

## Logarithmic functions

As long as b ≠ 1, the exponential function with base b is one-to-one, so it has an inverse. A logarithmic function is an inverse of an exponential function. These two statements mean exactly the same thing in the real-number system:
• bx = y, b > 0, and b ≠ 1;
• logby = x.
The left-hand side of the latter equation is the logarithm, base b, of y; logarithms are particularly useful in many applications of mathematics.

If you don't remember any other values of a logarithmic function, remember these:

• logb 1 = 0,
• logbb = 1,
• logb (1/b) = −1.
The domain of a logarithmic function is the set of all positive numbers; the range is the set of all real numbers. (A logarithm of a negative number is imaginary.) If b > 1, then the logarithmic function is increasing; if b < 1, then the logarithmic function is decreasing.

There are abbreviations for logarithms with certain special bases:

• lb x = log2x;
• lg x = log10x;
• ln x = logex, where e is the same special number from before, about 2.72;
• log x is the logarithm of x with whichever base is your favourite.
The textbook's favourite base is 10, so I will also use that. However, a lot of other people use e, and some people occasionally use 2. For this reason, ‘log’ without a subscript can be ambiguous, so the symbols ‘lb’, ‘lg’, and ‘ln’ are safer (and shorter).
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