The logarithm function and its properties

The logarithm as the inverse function of the exponential

In the lesson on the exponential function and powers with real exponents it was proved that, if $a > 0$ and $a \neq 1$, the function that assigns to each real number $x$ the power $a^x$ is strictly increasing if $a > 1$ and strictly decreasing if $0 < a < 1$; and, by the theorem on the range of the exponential function, proved there with the completeness of the reals and Bernoulli’s inequality, it moreover takes each positive value exactly once, and is therefore a bijection from $\mathbb{R}$ onto the interval $\ioo{0}{\infty}$. By the theorem on the inverse of a strictly monotonic function, proved in the lesson on function operations, this function has an inverse. If this is so, for each positive number $x$ there is a unique exponent $y$ such that $a^y = x$, and the definition below makes sense.

Let $a > 0$, with $a \neq 1$, and $x > 0$. The logarithm to base $a$ of $x$, written $\log_a x$, is the unique real number $y$ such that $a^y = x$. The function $\log_a$, defined on $\ioo{0}{\infty}$ and with real values, is the inverse of the exponential with base $a$.

The logarithm therefore answers the question of the exponent: $\log_a x$ is the number to which $a$ must be raised to obtain $x$. Thus, since $2^3 = 8$, $3^{-2} = \frac{1}{9}$, $10^3 = 1000$ and, by the lesson on powers with rational exponents and radicals, $4^{\frac{1}{2}} = \sqrt{4} = 2$, we have:

Since $\log_a$ and the exponential with base $a$ are inverses of each other, each undoes what the other does: the first identity holds for every positive $x$, and the second for every real $y$.

In particular, from $a^0 = 1$ and $a^1 = a$ follow $\log_a 1 = 0$ and $\log_a a = 1$ for every base. Neither zero nor the negative numbers have a logarithm, because every power with a positive base is positive: there is no $y$ such that $2^y = -4$. Two bases have names of their own. The base $e$, the number studied in the lesson on the number e and the natural exponential, gives the natural logarithm, written $\ln x$; the base $10$ gives the common logarithm, written $\log_{10} x$. Many texts and calculators write $\log x$ without a base, sometimes for the natural one and sometimes for the common one; in this course the base is always written.

Logarithms were published by John Napier in 1614, and a few years later Henry Briggs computed the first tables of common logarithms. For more than three centuries they served to shorten the computations of astronomy, navigation and engineering because, as is proved below, they turn products into sums. The graph of the logarithm is obtained from that of the exponential by the theorem on the graph of the inverse function, from the same lesson on the inverse function: the graph of a bijective function and that of its inverse are symmetric with respect to the line $y = x$.