### elevating a numbers to the power which is a positive totality number

The principle of logarithms arose from the of strength of numbers. If the nature of strength are familiar to you, friend may conveniently skim with the product below. If not--well, here are the details. Strength of a number are obtained by multiplying it by itself. Because that instance2.2 can be written 22 "Two squared" or "2 to the 2nd power"2.2.2 = 23 ."Two cubed" or "2 come the 3rd power""2.2.2.2 = 24 "Two to the fourth power" or merely "2 come the 4th"".2.2.2.2.2 = 25 "Two to the 5th power" or merely "2 come the 5th""2.2.2.2.2.2 = 26 "Two to the sixth power" or simply "2 come the 6th"" and also so on... The number in the superscript is recognized as one "exponent." The unique names because that "squared" and "cubed" come due to the fact that a square of side 2 has area 22 and a cube of side 2 has actually volume 23. Similarly, a square of next 16.3 has area (16.3)2 and a cube of side 9.25 has volume (9.25)3. Note the use of parentheses--they are not for sure needed, however they assist make clear what is increased to the second or 3rd power.Quick Quiz:The Greek Pythagoras confirmed (about 500 BC) that if (a,b,c) room lengths the the political parties of a right-angled triangle, with c the longest, climate a2 + b2 = c2In a ideal angles triangle, a = 12, b = 5. Have the right to you guess c? i beg your pardon is larger--23 or 32? 27 or 53?A slight change of an old riddle goes: together I to be going come St. IvesI met a guy with 7 wivesEach wife had seven sacksEach sack had seven catsEach cat had actually seven kitsKits, cats, man, wives--how plenty of were comes from St. Ives?It all requires powers that 7:Man -- 70 = 1Wives-- 71 = 7Sacks-- 72 = 49 (but they are not component of the count)Cats-- 73 = 343Kits-- 74 = 2401 complete count: 1 + 7 + 343 + 2401 = 2752As noted, this is contempt modified native the original riddle, i beg your pardon asks "how countless were going to St. Ives?" The prize is of course just one, the human being telling the riddle. Many listeners but are distracted by the numerous details given, miss the difference and also perform the over calculation. Their answer is climate wrong! The famed Indian mathematician Ramanujan to be sick in a hospital (tuberculosis, probably) as soon as he was saw by his friend the mathematician G.H Hardy, who had earlier invited him come England. Hardy later on told:I remember when going to see him as soon as he was ill in ~ Putney. I had ridden in taxi cab number 1729 and remarked the the number seemed to me rather a dull one, and also that i hoped it was not negative omen. "No," that replied, "it is a an extremely interesting number; that is the smallest number expressible together the amount of 2 cubes in two different ways."Cubes are 3rd powers. What are they, in this example? shot guessing, choices are limited.You are watching: 6 to the 5th power

### Multiplying powers

Note that (23).(22) = 25since the very first term contributes three determinants of 2 and also the 2nd term contributes two--together, 5 multiplications by 2. The exact same will organize if "2" is replaced by any type of number. So, if the number is stood for by "x" we get (x3).(x2) = x5and in basic (since there is nothing special around 2 and 3 which will certainly not hold for other totality numbers) (xa).(xb) = x(a+b)where a and also b are any type of whole numbers. The most widely offered powers by entirety numbers, for customers of the decimal system, room of course those of 10101 = 10 ("ten")102 = 100 ("hundred")103 = 1000 ("thousand")104 = 10,000 ("ten thousand"))105 = 100,000 ("a hundred thousand")106 = 1,000,000 ("a million") keep in mind that here the "power index" also gives the variety of zeros. For larger numbers, it used to be the in the united state 109 = 1,000,000,000 was referred to as "a billion" while in Europe it was referred to as a "milliard" and also one had to advance to 1012 to with a "billion." these days the us convention is getting ground, however the civilization remains divided between nations wherein the comma denotes what we contact "the decimal point", while the suggest divides huge numbers, e.g. 109 = 1.000.000.000 (in the united state commas would be used). It also should be noted that some societies have assigned names to some various other powers the 10--e.g. The Greeks used "myriad" for 10,000 while the Hebrew holy bible named that "r"vavah," and in India "Lakh" still means 100,000, if "crore" is 10,000,000. A 9-year old in 1920 coined the name "Googol" because that 10100, yet the word found little use beyond inspiring the surname of a search engine on the world-wide web.### Dividing one power by another

In a path very comparable to the above, we deserve to write (25) / (22) = 23since dividing a power of 2 by some smaller sized power way canceling indigenous the molecule a number of factors equal to those in the denominator. Writing it the end in detail(2.2.2.2.2) / (2.2) = 2.2.2 right here too the number elevated to greater power need not be 2--again, signify it by x--and the powers require not it is in 5 and 2, however can be any kind of two totality numbers, to speak a and also b: (xa) / (xb) = x(a–b) Here but a brand-new twist is added, due to the fact that subtraction can additionally yield zero, or even negative numbers. Before exploring the direction, that helps outline a general course come follow.### Expanding the meaning of "Number"

earlier at the dim beginnings of humanity, "numbers" just meant positive totality numbers ("integers"): one apple, two apples, 3 apples... Straightforward fractions were also found useful--1/2, 1/3 and also so on.Then zero to be added, initially from India.Then an unfavorable numbers were given complete status--rather than check out subtraction as a separate operation, it was re-interpreted as addition of a an unfavorable number. Similarly, come every creature x there corresponded an "inverse" number (1/x) (many calculators have a 1/x switch too). In old Egypt, 5000 year ago, these were the only fractions recognized, and they are therefore still sometimes dubbed "Egyptian fractions." once an Egyptian of the time want to refer 3/4, it was presented as (1/2 + 1/4). Sometimes long expressions were needed, e.g 99/100 = 1/2 + 1/4 + 1/5 + 1/25but it constantly worked. The ancient Greeks went further and also defined together "rational number" (or "logical" numbers--"rational" comes from Latin) any type of multiple of such an inverse, for instance 4/13, 22/7 or 355/113. Reasonable numbers room dense: no matter how close 2 of them room to each other, one could constantly place another rational number between them--for instance, fifty percent their amount is one selection out of many. Decimal fountain which avoid at some length are rational number too, despite decimal fractions having actually infinite length however with a repeating pattern (0.33333..., 0.575757... Etc.) can always be expressed together rational fractions. Greek theorists in the early days of math were therefore surprised to uncover that regardless of that density, some extra numbers could still "hide" between rational ones, and also could no be stood for by any kind of rational number. Because that instance, √2 is that this class, the number who square equates to 2. Many square roots and solutions that equations are additionally of this kind, together is π, the ratio between the one of a circle and its diameter (denoted by the Greek letter "pi"). Pi has a fair approximation in 22/7 and also a much much better one in 355/113, however its exact value can never be stood for by any fraction. Mathematicians check out all the preceding types of number together a single class of "real numbers". Logarithms of hopeful numbers are actual numbers, too. When one writes2 = 100.3010299.. so that 0.3010299.. = log 2(the dots stand for an rarely often, rarely continuation) one views it together 10 elevated to a power which is some actual number. Earlier, powers were integers, denoting the number of times some number to be multiplied by itself. To do the over expression meaningful, it is because of this necessary to generalize the principle of elevating a number to some power to where any real number can be the strength index.### Logarithms of powers of 10

These are all whole numbers:101 = 10 so log 10 = 1102 = 100 so log 100 = 2 103 = 1000 so log in 1000 = 3 104 = 10,000 so log10,000 = 4 105 = 100,000 so log 100,000 = 5 106 = 1,000,000 so log in 1,000,000 = 6 this logarithms additionally satisfy the rules we found(xa).(xb) = x(a+b)So if x=10 U = (10a) V = (10b) W = (10(a+b)) = U.Vthen due to the fact that a = log in U b = logV (a+b) = log Wwe have logV + log U = log in (U.V) This relation holds anytime U and V room powers that 10: The logarithm the the product is the amount of the logarithms of the multiply numbers. as demonstrated in the rundown in the coming before section. Together the ide of logarithm is broadened, the property constantly remains. The is what originally made logarithms useful: convert multiplication right into addition. Rather of having to multiply U and V, we just need add their logarithms and also then look because that the number whose logarithm equals that sum: that will be the product (U.V).Similarly, (xa) / (xb) = x(a–b)so if x=10, U = (10a) V = (10b) W = (10(a–b)) = U/Vthen in the department we havelogU – log V = log (U/V) or "the logarithm the the quotient is the difference in between the logarithms that the split numbers," e.g. 107 / 104 = 103 i m sorry agrees v 7 – 4 = 3.Division, though, opens up up a new territory: by the very same rule, for circumstances 1040 / 1043 = 10–3 = 0.001And 104 / 104 = 100 = 1 because a number is being split by itself must equal 1.Indeed, this is continual with the rule, the including or subtracting 1 come the logarithm move its number one decimal to the appropriate of left. Earlier106 = 1,000,000 so log 1,000,000 = 6 105 = 100,000 so log 100,000 = 5 104 = 10,000 for this reason log10,000 = 4 103 = 1000 so log in 1000 = 3 102 = 100 so log 100 = 2 101 = 10 so log 10 = 1and now this can be extended, splitting by 10 at every step100 = 1 so log in 1 = 010–1 = 0.1 so log 0.1 = –1 10–2 = 0.01 so log 0.01 = –2 10–3 = 0.001 so log in 0.001 = –3 10– 4 = 0.000 1 so log in 0.000 1 = –4 10–5 = 0.000 01 so log 0.000 01 = –5 10–6 = 0.000 001 so log 0.000 001 = –6 The above demonstrates an additional property the logarithms:Log (VQ) = Q log in VFor the special instance V = 10, logV = 1## Scientific Notation

The amounts with i m sorry scientists work-related are sometimes very tiny or really large. That is then practically (for calculation, and also for using logarithms) to separate the number into two parts--a number from 1 to 10, giving its structure, and also a strength of 10, providing the magnitude. Electrical charge, for instance, is measured in coulombs: about one coulomb operation each second through a 100-watt lightbulb. That current is lugged by a huge variety of tiny an unfavorable particles, uncovered in any kind of atom and also known as electrons. Every electron tote a charge of q = 1.60219 10–19 coulomb If this to be to be created as a decimal fraction, the expression would take about half a line, with 18 zeros complying with the decimal suggest in front of the far-reaching digits--and a rapid look in ~ it would not give much information, one still would have had to counting the zeros. The mass of the electron is likewise smallm = 9.1095 10–29 kgScientific notation simplifies composing such numbers. Yet another example is the speed of light, as decimal number (accuracy come 6 figures) 299,792,000, in scientific notation c = 2.99792 108 meter/second clinical notation additionally makes multiplication and department easier and less error prone. One multiplies or divides individually the numerical factors, each between 1 and also 10, and also usually sees at a glance if the an outcome is the the right range of magnitude. Separately, one adds with each other all power exponents of multiplied factors, and subtracts those of divided ones, to get the appropriate power that 10 i beg your pardon then shows up in clinical notation. Of course, in any type of calculation, one must use regular units--it would certainly not execute to mix meters and inches, or pounds and also grams (such inconsistent usage apparently led to an error which resulted in a room probe to Mars to miss the planet and get lost). The most common consistent system in physics and an innovation is the MKS system, measuring distance in meters, fixed in kilograms and time in seconds. All other units are figured out by the choice of these three standards, and as long as one stays in the MKS system, outcomes conform to devices of that device too (e.g. If power is gift calculated, it constantly comes the end in joules).### An example

electron of the polar aurora ("northern lights") move at about 1/5 the velocity that light, in a magnetic ar B which near the ground is about 5 10–5 Tesla (the Tesla is the MKS unit of magnetic field: in ~ the pole of an steel magnet girlfriend get around 1 Tesla). The magnetic field reasons an electron come spiral roughly the direction that the magnetic force ("magnetic ar line") with a radius the r = mv/(qB)where v is the component of the velocity perpendicular come the direction of B. If the component perpendicular to B is fifty percent the full velocity (i.e c/10), what is r?We have m = 9.1095 10–29 Kg v = 2.99791 107 m/sec (= 0.1 c) q = 1.60219 10–19 coulomb B = 5 10–5 TeslaCollecting every numerical factors, and rounding off to 3 decimals(9.11).(3.00)/<(1.6).(5)> = 3.42Collecting every exponents(– 29+7) – (–19 – 5) = (– 22) – (– 24) = +2The radius is therefore 3.42 102 meter or 342 meters. That is the stimulate of the radius the a an extremely thin auroral ray, watched from the ground. Because the ground native which aurora is usually viewed is 100 kilometers listed below the aurora, together a beam must appear as really thin indeed.Next section: (M-15) raising one power to Another and a graphic use of logarithms.Back to the understand List**Timeline**

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David P. SternMail come Dr.Stern: stargaze("at" symbol)londonchinatown.org .Updated 9 November 2007, edited 28 October 2016