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 = 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 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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