Logarithm

Did You Know?

• The logarithm of the base itself is always 1, because b^1 = b.
• Logarithms can be negative; for example, log_10(0.01) = -2.

Analytic properties

The logarithm function is defined as the inverse of the exponential function f(x) = b^x, where b is a positive real number not equal to 1. Because f(x) = b^x is continuous and strictly monotonic (increasing for b > 1, decreasing for b < 1) on its domain of all real numbers, it is bijective onto the positive reals. Thus, for each positive real number x, there is exactly one real number y such that b^y = x, and this y is denoted log_b(x). The function log_b(x) can also be characterized as the only increasing function from the positive reals to the reals that satisfies f(b) = 1 and the product formula f(xy) = f(x) + f(y). The graph of the logarithm is obtained by reflecting the graph of the exponential across the line y = x. As x grows to infinity, log_b(x) diverges to infinity for b > 1 and to negative infinity for b < 1. The logarithm is continuous and differentiable wherever it is defined; its derivative is 1/(x ln(b)).

Calculation

Some logarithms are straightforward, such as log_10(1000) = 3. In general, logarithms can be computed using power series, the arithmetic–geometric mean, or precomputed logarithm tables. Newton's method can also be used, since the exponential function can be computed efficiently. CORDIC-like methods compute logarithms using only addition and bit shifts, while the binary logarithm algorithm recursively squares x. The natural logarithm ln(z) for 0 < z ≤ 2 can be approximated by the Taylor series ln(1+z) = z - z^2/2 + z^3/3 - ..., which converges quickly for small z. A faster-converging series for ln(z) uses the inverse hyperbolic tangent function: ln(z) = 2 * sum_{k=0}^∞ (1/(2k+1)) * ((z-1)/(z+1))^(2k+1). For z close to 1, this series converges rapidly. A low-accuracy approximation y ≈ ln(z) can be refined by computing A = z / e^y, then ln(z) = y + ln(A), where ln(A) can be calculated efficiently if y is not too large. Larger values of z can be reduced by writing z = a * 10^b.

Applications

Logarithms appear in many contexts related to scale invariance and self-similarity. The shell of a nautilus exhibits a logarithmic spiral, and Benford's law on leading digits is explained by scale invariance. In algorithm analysis, logarithms arise when a problem is divided into two similar smaller subproblems. The dimensions of self-similar geometric shapes, such as fractals, are based on logarithms. Logarithmic scales compress large-scale data and quantify relative change rather than absolute difference. Scientific formulas incorporating logarithms include the Tsiolkovsky rocket equation, the Fenske equation, and the Nernst equation. The decibel, a logarithmic unit, measures ratios of power or amplitude, commonly used for sound levels, electrical signal attenuation, and light absorbance. The signal-to-noise ratio and peak signal-to-noise ratio, used in assessing sound and image compression, are also expressed in decibels.

Generalizations

The logarithm can be extended to complex numbers. For a non-zero complex number z, represented in polar form as z = r e^(iφ) (where r is the positive real absolute value and φ is the argument), the complex logarithm is multi-valued. Because adding 2πk (for any integer k) to φ yields the same complex number, the logarithm of z is given by ln(r) + i(φ + 2πk) for all integers k. The principal value, denoted Log(z), selects the argument within a chosen interval, such as (-π, π] or [0, 2π). For a positive real number x, the principal argument is 0, so Log(x) equals the real natural logarithm. However, the usual identities for logarithms of products and powers do not hold for the principal value of the complex logarithm. The discrete logarithm, defined as the multi-valued inverse of exponentiation in finite groups, is used in public-key cryptography.