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:
The left-hand side of the latter equation
is the logarithm, base b, of y;
logarithms are particularly useful in many applications of mathematics.
- bx = y,
b > 0, and b ≠ 1;
- logb y = x.
If you don't remember any other values of a logarithmic function,
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.
- logb 1 = 0,
- logb b = 1,
- logb (1/b) = −1.
There are abbreviations for logarithms with certain special bases:
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).
- lb x = log2 x;
- lg x = log10 x;
- ln x = loge 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.
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