For example, since we know that 10^3 = 1000, we also know immediately that the base-10 logarithm of 1000 is 3 (because 3 is the power that you need to raise 10 to, to get 1000). It is thus trivial to know the base-10 log of numbers such as 1, 10, 100, 1000, etc., because these numbers are integer powers of the base and we recognize them as such. The algorithms used in computers typically use iterative numerical methods to obtain highly accurate results, and may use either a lookup table or a crude formula to obtain an initial approximation.)Īlthough generally it is no simple task to obtain the logarithm of a given number, it is trivial to obtain a logarithm when the given number (the argument) happens to be an integer power of the base of the logarithm (and when we are familiar with the base and with the integer powers of the base). (Or else approximated by summing the first several terms of a series expansion of the function, or by using numerical methods that involve iteration. Prior to the proliferation of computers and calculators, it was ordinarily done using a lookup table or else a slide rule. One consequence of the abstract way in which logarithms are defined is that there isn’t any straightforward way to calculate a logarithm. No matter the base, any exponential function implicitly has an inverse, known as a logarithm function. In a nutshell, this is what an exponential function is, however we should note that even though we are mainly concerned with base 10, that in general the base of an exponential function can be any constant. So what exactly is an exponential function? Suppose you are given some positive integer number N, and you raise the constant number 10 to the power N (treating N as the independent variable). Logarithms are defined by way of inverse functions for exponential functions. ![]() This behavior is exemplified by the familiar calculator function ‘1/X’, which function is its own inverse. Not all mathematical functions have an inverse function, but if a function has an inverse function, the two functions are complementary in the sense that each reverses the effect of the other. Logarithms arise from the concept of the inverse function. Whenever I use the word “physical”, this is my meaning. Occasionally in this article I will need to make some point by saying something about the corresponding physical value, i.e., the value that is accompanied by physical units of measure. ![]() When decibels are used to express signal-to-noise ratios (which are always expressed in decibels), the dimensionless ratio is the ratio of the strength of the signal to the strength of the noise included in the signal. When decibels are used to express dynamic range (the dynamic range of the Compact Disc format for example), the dimensionless ratio is the ratio of the loudest signal that can be encoded in the format, to the quietest non-zero signal that can be encoded in the format. This isn’t always the case, and when it isn’t the case, the numerator and denominator will still agree on the physical units of measure. Whenever power, voltage, etc., is converted to a dimensionless ratio (the first step in translation to decibel), the denominator in the division operation will often be a standard reference value such as one milliWatt or one milliVolt. This effect is prominent, however the underlying rationale is not obvious and has to do with the mathematical properties of logarithms. Logarithmic scaling pushes large values closer together and spreads small values further apart. ![]() (Physicists regard all units of measure as “dimensions”, hence a value that is not accompanied by physical units of measure is “dimensionless”.)įact #2: Decibel scaling is logarithmic. When the given quantity is divided by the reference value, the physical units of measure vanish. Relative density is obtained by dividing the density of a given substance by the density of another substance used as a reference, usually water. Mach numbers are obtained by dividing a given speed by the speed of sound, which is used as a reference. Two familiar examples of dimensionless ratios are Mach numbers and relative density. Let’s begin with two essential facts about decibels:įact #1: Decibel values are a type of “dimensionless ratio”. Each of these different physical quantities is either a form of power or else has a very specific and simple relationship with power. Decibels are used routinely for expressing electrical power, electrical voltage, sound power and (more commonly) sound pressure level (SPL).
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