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) = b^x,

for some constant b called the base of the function. The base should be a positive number, so that b^x 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) = Cb^x,

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, b^x 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 e^x 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:

b^x = y, b > 0, and b ≠ 1;
log_b y = 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:

log_b 1 = 0,
log_b b = 1,
log_b (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 = log₂ x;
lg x = log₁₀ x;
ln x = log_e x, 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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