Logarithmic functions (§6.4)

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.

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

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).
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