# Logarithmic functions (§5.4)

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* = log_{2} *x*;
- lg
*x* = log_{10} *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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