Exponential and logarithmic functions (§§6.3&6.4)

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

Exponential functions

Recall that a power function is a function f of the form

f(x) = xⁿ,

for some constant n called the exponent of the function. 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.

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 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 exponential 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:

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 whatever is your favourite base.

The book's favourite base is 10, which I will also use.

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