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

A power function is a function f of the form for some constant n called the exponent of the function. You already know about power functions.

An exponential function is a function f of the form

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.

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

The domain of an exponential function is the set of all real numbers; as long as b ≠ 1, the range is the set of all positive numbers. (Because b is positive, bx is also positive.) If b > 1, then the exponential function is increasing; if b < 1, then the exponential function is decreasing; and if b = 1, then the exponential function is constant.

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:

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

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 exponential function is increasing; if b < 1, then the exponential function is decreasing.

There are abbreviations for logarithms with certain special bases:

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