Logarithm

The logarithm of $x$ to the base $b$, written $\log_b x$, is the exponent $y$ with $b^y = x$: how many times $b$ must be multiplied together to give $x$. It exists for $x > 0$, with $b > 0$ and $b \ne 1$. The common bases are 10 ($\log_{10}$, the common logarithm), $e$ ($\ln$, the natural logarithm) and 2. This page explains the curve $y = \log_b x$ by one rule.

On the x axis a chain of points starts at $x_0$ and is multiplied by the same $m$ again and again: $x_0, x_0 m, x_0 m^2, \dots$, and divided by it the other way. For $m > 1$ the points spread farther and farther apart along the x axis (for $m < 1$ the chain runs the other way), yet on the curve every step has the same height $\log_b m$, because $\log_b(xy) = \log_b x + \log_b y$: multiplying the numbers adds their logarithms. The yellow notches on the y axis are evenly spaced. Starting from $x_0 = 1$ with $m = b$, the heights are 0, 1, 2, 3: the logarithm counts the factors of $b$. With $m = \sqrt b$ two steps make one factor of $b$, so each is ½ high — that is where fractional exponents come from. The rule shapes the whole curve: it passes through $(1, 0)$ and $(b, 1)$; it flattens, since one more step up needs the same factor, a longer and longer stretch of the x axis; and dividing towards 0 never reaches it, so for $b > 1$ the curve falls without bound as $x \to 0^+$.

On a log scale a number sits at the position $\log x$. The chain then becomes evenly spaced and the curve a straight line: equal ratios are equal distances. A slide rule is two such scales: sliding one along by $\log x_0$ multiplies every number on it by $x_0$, and you read products by eye to about three figures. Changing the base only rescales the heights, $\log_b x = \ln x / \ln b$; for $0 < b < 1$ the curve is turned upside down, $\log_{1/b} x = -\log_b x$. And $\log_b(kx) = \log_b x + \log_b k$: scaling the curve sideways by $1/k$ is the same as moving it up by $\log_b k$.