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:

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).

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2023 September 12. Toby reserves no legal rights to it.

The permanent URI of this web page is https://tobybartels.name/MATH-1150/2024FA/logs/.