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logarithm of x”. In symbols, y = logb(x). Every exponential equation can be rewritten as a logarithmicequation, and vice versa, just by interchanging the x and y in this way. Another way to look at it is that the logbx function is de ined as the inverse of the bx function. Thesetwo statements express that inverse relationship, showing how an exponential equation is equivalentto a logarithmic equation: x = by is the same as y = logbxExample 1: 1000 = 103 is the same as 3 = log101000.Example 2: log381 = ? is the same as 3? = 81.It can’t be said too often: a logarithm is nothing more than an exponent. You can write the abovede inition compactly, and show the log as an exponent, by substituting the second equation into the irstto eliminate y:Read that as “the logarithm of x in base b is the exponent you put on b to get x as a result.” Where Did Logs Come From?Before pocket calculators—only three decades ago, but in “student years” that’s the age ofdinosaurs—the answer was simple. You needed logs to compute most powers and roots with fairaccuracy; even multiplying and dividing most numbers were easier with logs. Every decent algebrabooks had pages and pages of log tables at the back. The invention of logs in the early 1600s fueled the scienti ic revolution. Back then scientists,astronomers especially, used to spend huge amounts of time crunching numbers on paper. By cuttingthe time they spent doing arithmetic, logarithms effectively gave them a longer productive life. Theslide rule, once almost a cartoon trademark of a scientist, was nothing more than a device built fordoing various computations quickly, using logarithms. See Eli Maor’s e: The Story of a Number for moreon this. Today, logs are no longer used in routine number crunching. But there are still good reasons forstudying them. Why Do We Care?Why do we use logarithms, anyway? I could write a whole article about them—maybe one day. But fornow. ... To ind the number of payments on a loan or the time to reach an investment goal. To model many natural processes, particularly in living systems. We perceive loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale. We also perceive brightness of light as the logarithm of the actual light energy, and star magnitudes are measured on a logarithmic scale. 2 of 10
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To measure the pH or acidity of a chemical solution. The pH is the negative logarithm of the concentration of free hydrogen ions. To measure earthquake intensity on the Richter scale. To analyze exponential processes. Because the log function is the inverse of the exponential function, we often analyze an exponential curve by means of logarithms. Plotting a set of measured points on “log‐log” or “semi‐log” paper can reveal such relationships easily. Applications include cooling of a dead body, growth of bacteria, and decay of a radioactive isotopes. The spread of an epidemic in a population often follows a modi ied logarithmic curve called a “logistic”. To solve some forms of area problems in calculus. (The area under the curve 1/x, between x=1 and x=A, equals ln A.) Also in calculus, differentiating a complicated product becomes much easier if you irst take the logarithm.(Historically, the main reason for teaching logs in grade school was to simplify computation, becausethe log of a multiplication “downgrades” it to an addition, and the log of a power expression“downgrades” it to a multiplication. Of course, with the widespread availability of personal computingdevices, dif iculty of computation is no longer a concern, but logs still have many applications in theirown right.)“Base”ic FactsFrom the de inition of a log as inverse of an exponential, you can immediately get some basic facts. Forinstance, if you graph y=10x (or the exponential with any other positive base), you see that its range ispositive reals; therefore the domain of y=log x (to any base) is the positive reals. In other words, youcan’t take log 0 or log of a negative number. (Actually, if you’re willing to go outside the reals, you can take the log of a negative number. Thetechnique is taught in many trigonometry courses.) Log of 1, Log Equaling 1You know that anything to the zero power is 1: b0 = 1. Change that to logarithmic form with thede inition of logs and you have logb1 = 0 for any base bIn the same way, you know that the irst power of any number is just that number: b1 = b. Again, turnthat around to logarithmic form and you have logbb = 1 for any base bExample 3: ln 1 = 0 3 of 10
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Example 4: log55 = 1 Log as InverseA log is an exponent because the log function is the inverse of the exponential function. The inversefunction undoes the effect of the original function. (I’m not a big fan of most uses the term “cancel” inmath, but it does it in this situation.)This means that if you take the log of an exponential (to the same base, of course), you get back towhere you started: logbbx = x for any base bThis fact lets you evaluate many logarithms without a calculator.Example 5: log5125 = log5(5³) = 3Example 6: log10103.16 = 3.16Example 7: ln e-kt/2 = -kt/2 What’s “ln”?Any positive number is suitable as the base of logarithms, but two bases are used more than any others: base of symbol name logarithms log 10 common logarithm (if no base shown) natural logarithm, e ln pronounced “ell‐enn” or “lahn”Natural logs are logs, and follow all the same rules as any other logarithm. Just remember ln x means logexWhy base e? What’s so special about e? Most of the explanations need some calculus, for instance that exis the only function that is both its own integral and its own derivative or that e has this beautifulde inition in terms of factorials: e = 1/0! + 1/1! + 1/2! + 1/3! + ...Numerically, e is about 2.7182818284. It’s irrational (the decimal expansion never ends and neverrepeats), and in fact like pi it’s transcendental (no polynomial equation with integer coef icients has pior e as a root.) e (like pi) crops up in all sorts of unlikely places, like computations of compound interest. It wouldtake a book to explain, and fortunately there is a book, Eli Maor’s e: The Story of a Number. He also goes 4 of 10
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into the history of logarithms, and the book is well worth getting from your library.Combining Logs with the Same BaseIn a minute we’ll look at the various combinations. But irst you might want to know the generalprinciple: logs reduce operations by one level. Logs turn a multiplication into an addition, a division intoa subtraction, an exponent into a multiplication, and a radical into a division. Now let’s see why, andlook at some examples. Multiply Numbers, Add Their LogarithmsMultiplying two expressions corresponds to adding their logarithms. Can we make sense of this?By the compact de inition, x = blogbx and y = b b log yand therefore, substituting for x and y, xy = blogbx blogbyBut when you multiply two powers of the same base, you add their exponents. So the right‐hand sidebecomes xy = blogbx+logbyNow apply the compact de inition to the left=hand side: blogb(xy) = xyCombine that with the preceding equation to obtain blogb(xy) = blogbx+logbyNow we have two powers of the same base. If the powers are equal, then the exponents must also beequal. Therefore logb(xy) = logbx + logbySo what’s the bottom line? Multiplying two numbers and taking the log is the same as taking their logsand adding.Example 8: log8(x)+log8(x²) is the same as log8(xx²) or just log8(x³).Example 9: log10(20)+log10(50) = log10(20×50) = log10(1000) = 3. Exponent, Multiply the LogarithmContinuing our theme of logarithms reducing the level of operations, if you have the yth power of anumber and take the log, the result is y times the log of the number. Here’s why, starting with xy: 5 of 10
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Start with the compact de inition of a logarithm: x = blogbxand raise both sides to the y power: xy = (blogbx)yA power of a power is equivalent to just multiplying the exponents. Simplify the right‐hand side: xy = b(y logbx)Rewrite the left‐hand side using the compact de inition of a log: y blogb(x ) = xy(The font may be hard to read: that’s x to the power y on left and right.) and combine the last twoequations: y blogb(x ) = b(y logbx)If the powers are equal and the bases are equal, the exponents must be equal: logb(xy) = y logbxExample 10: ln(26) = 6 ln 2 (where “ln” means loge, the natural logarithm).Example 11: log5(5x²) is not equal to 2 log5(5x). Be careful with order of operations! 5x² is 5(x²), not(5x)². log5(5x²) must irst be decomposed as the log of the product: log55 + log5(x²). Then the secondterm can use the power rule, log5(x²) = 2 log5x. The irst term is just 1. Summing up, log5(5x²) = 1 +2 log5x. Raising Numbers to Any PowerThe trick to evaluating expressions like 6.74.4 is to use the exponent rule and the log‐as‐inversede inition: x = 6.74.4 log x = 4.4 ( log 6.7 ) = about 3.634729132 x = 103.63472... = about 4312.5There’s nothing special about base‐10 logs here. The calculation could just as well be x = 6.74.4 ln x = 4.4 ( ln 6.7 ) = about 8.369273116 x = e8.36927... = about 4312.5This will work for any positive base and any real exponent, so for example x = ππ log x = π (log π) = about 1.561842388 x = 101.5618... = about 36.46215961You can combine this with the multiplying numbers = adding logarithms rule to evaluate powers thatare too big for your calculator. For example, what is 671217? x = 671217 log x = 217 (log 671) = about 613.3987869Now, separate the integer and fractional parts of the logarithm. 6 of 10
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log x = about 0.3987869 + 613 x = 100.3987869 + 613 x = 100.3987869 · 10613 x = about 2.505 · 10613For examples like this, you really do have to use base‐10 logs.If the base is negative or the exponent is complex, see Powers and Roots of a Complex Number. Divide Numbers, Subtract Their LogarithmsSince division is the opposite of multiplication, and subtraction is the opposite of addition, it’s notsurprising that dividing two expressions corresponds to subtracting their logs. While we could go backagain to the compact de inition, it’s probably easier to use the two preceding properties.Start with the fact that 1/y = y−1 (see the de inition of negative exponents): x/y = x(1/y) = xy−1and take the log of both sides: logb(x/y) = logb(xy−1)The right‐hand side is the log of a product, which becomes the sum of the logs: logb(x/y) = logbx + logb(y−1)and the second term is the log of a power, which becomes (−1) times the log, or just minus the log: logb(x/y) = logbx − logbyIn words, if you divide and take the log, that’s the same as subtracting the individual logs.Example 12: 675÷15=45, and therefore log10675 − log1015 = log1045. (Try it on your calculator!)Example 13: log(x³y²) − log(x²y³) = log(x³y² / x²y³) = log(x/y) = log(x) − log(y).Changing the BaseNow you have everything you need to change logarithms from one base to another. Look again at thecompact equation that de ines a log in base b:To change the log from base b to another base (call it a), you want to ind loga(x). Since you alreadyhave x on one side of the above equation, it seems like a good start is to take the base‐a log of bothsides: loga(blogbx) = logax 7 of 10
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But the left‐hand side of that equation is just the log of a power. You remember that log(xy) is just log(x)times y. So the equation simpli ies to (logab) (logbx) = logaxNotice that logab is a constant. This means that the logs of all numbers in a given base a areproportional to the logs of the same numbers in another base b, and the proportionality constant logabis the log of one base in the other base. If you’re like me, you may have trouble remembering whetherto multiply or divide. If so, just derive the equation—as you see, it takes only two steps.Some textbooks present the change‐of‐base formula as a fraction. To get the fraction from the aboveequation, simply divide by the proportionality constant logab: logbx = (logax) / (logab)Example 14: log416 = (log 16) / (log 4). (You can verify this with your calculator, since you knowlog416 must equal 2.)Example 15: Most calculators can’t graph y = log3x directly. But you can change the base to e and easilyplot y = (ln x)÷(ln 3). (You could equally well use base 10.)An interesting side road leads from the above formula. Replace x everywhere with a—this is legal sincethe formula is true for all positive a, b, and x. You get logba = (logaa) / (logab)But logaa = 1 (see Log of 1 above), so the formula becomes logba = 1 / (logab)Example 16: log10e = 1/(ln 10). (You can verify this with your calculator.)Example 17: log1255 = 1/(log5125). This is easy to verify: 53 = 125, and 5 is the cube root of 125.Therefore log1255 = 1/3 and log5125 = 3, and 1/3 does indeed equal 1/3.SummaryThe laws of logarithms have been scattered through this longish page, so it might be helpful to collectthem in one place. To make this even more amazingly helpful <grin>, the associated laws of exponentsare shown here too. For heaven’s sake, don’t try to memorize this table! Just use it to jog your memory as needed. Betteryet, since a log is an exponent, use the laws of exponents to re‐derive any property of logarithms thatyou may have forgotten. That way you’ll truly gain mastery of this material, and you’ll feel con identabout the operations. exponents logarithms 8 of 10
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(All laws apply for any positive a, b, x, and y.) x = by is the same as y = logbx b0 = 1 logb1 = 0 b1 = b logbb = 1 b(logbx) = x logbbx = x bx by = bx+y logb(xy) = logbx + logby bx÷by = bx−y logb(x/y) = logbx − logby (bx)y = bxy logb(xy) = y logbx (logab) (logbx) = logax logbx = (logax) / (logab) logba = 1 / (logab)Don’t get creative! Most variations on the above are not valid.Example 18: log (5+x) is not the same as log 5 + log x. As you know, log 5 + log x = log(5x), not log(5+x).Look carefully at the above table and you’ll see that there’s nothing you can do to split up log(x+y) orlog(x−y).Example 19: (log x) / (log y) is not the same as log(x/y). In fact, when you divide two logs to the samebase, you’re working the change‐of‐base formula backward. Though it’s not often useful, (log x) /(log y) = logyx. Just don’t write log(x/y)!Example 20: (log 5)(log x) is not the same as log(5x). You know that log(5x) is log 5 + log x. There’sreally not much you can do with the product of two logs when they have the same base.See also: Combining Operations (Distributive Laws)ConclusionWell, there you have it: the laws of logarithms demysti ied! The general rule is that logs simply drop anoperation down one level: exponents become multipliers, divisions become subtractions, and so on. If 9 of 10
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ever you’re unsure of an operation, like how to change base, work it out by using the de inition of a logand applying the laws of exponents, and you won’t go wrong.What’s New 6 Feb 2012: Add applications here and here. 6 Mar 2011: Add Raising Numbers to Any Power (intervening changes suppressed) 11 Jan 1998: Adapt this article for the Web 22 Dec 1997: Post to alt.algebra.helpthis page: http://oakroadsystems.com/math/loglaws.htm 10 of 10
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