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Exponents and Logarithms In math, every operation has another operation that allows you to work backwards—addition has subtraction, multiplication has division and exponents have logarithms. Logarithms are used to determine what exponent a given base would require in order to produce a certain number. For further clarification assume the following generic expression for exponents: b x y Equation 1 where, b is called the “base”, x is the exponent and y is that certain number. With that in mind, we have the following as a generic expression for logarithm: log b y x Equation 2 where, all the variables are given the same label as above (b is base, x is exponent and y is that certain number). b x y bx y base exponent log b y x base exponent that “certain number” that “certain number” Using those first two equations (Equation 1 and 2) you can substitute x in Equation 1 with the statement in Equation 2 to find this rule: blogb y y Equation 3 Notice how the subscript of “log” is the same as the base? If both of those are equal, then that expression will always hold true. This becomes important later on. Examples using blogb y y 1. 4log4 17 17 2. 10 log21 21 3. eln100 100 No calculators were needed to figure those out. No real thought. All you do is realize you are in the form of blogb y y and then use that to your advantage. You may notice that Examples 2 and 3 may look weird. “What is ‘ln’?” and “Why isn’t there a b by the log?” Well, let me explain. There are two special logarithms that are used so frequently they’ve been given special notation. When you simply see “log” it is assumed to be of base 10 (our favorite of all the bases). When you see “ln” that is simply a “log” with a base of e (log e) which is 1 also known as “natural log”. Don’t know what “e” is? Check out the “What is e?” note below. When dealing with logarithms there are eight (8) special rules (or properties) that will make solving logarithmic problems easier. The rules are assuming M, N and b are all positive, non-zero numbers—there is no way to get the log of a negative number or zero. For example, log 2 4 is not possible. Feel free to grab your calculator if you don’t believe me. Rules 1. log 1 0 b 3. log b b x 5. log b MN log b M 7. log b M p p log b M x log b N 2. log b b 1 4. b logb x 6. M N 8. If log x b x y log b x log b M log b N log b y , then An important thing to remember is this works both left to right and right to left. In other words, log b M log b N log b MN is just as legitimate as log b MN log b M log b N ). What is e? It is a special number, like π (3.1415…), called a transcendental number. A transcendental number is an irrational number (a number that can’t be written into a fraction) and is not a solution of a non-zero polynomial. Do you need to know this to work these problems? No. All you need to know is that e (which is 2.71828…) represents a number (just like π). ex When x = x=0 x=1 x=2 x e = e0 =1 e1 =2.71828… e2 =7.38906… Examples using the aforementioned rules: 1. log 2 33 log 2 11 log 2 3 2. log 5 log 10 log 2 (because 10 divided by 2 is 5—I could have just as easily did log 20 log 4 or any other combination of numbers that, when divided, gives me a quotient of 5) 4 log 3 4 log 3 7 3. log 3 7 4. log 6 x 2 2 log x 5. log 2 16 4 log 2 2 4(1) equals 1) 4 (remember log b b always Some of these you can cleverly manipulate to use different rules. Who’s to say you need to use 11 and 3 for Example 1? You could have done log 2 33 log 2 99 log 2 3 where 99 3 33 just as easily. One last concept to have down and we should be good to go! What if you have something like log 2 7 ? Remember, that’s asking “Two (2) to what power is 7?” Well, there’s something you can do to make it easier: Change of Base. To do change of base, all you do is follow this formula: 2 log a x log a b Typically, we use a = 10 because our scientific calculators have a button conveniently placed on them (remember 10 is the favorite of all the bases). log a x Examples using log b x log a b log 7 1. log 4 7 1.4037 log 4 log 16 2. log 5 16 1.7227 log 5 ln 3 1.5850 3. log 2 3 ln 2 For Example 3, remember the base doesn’t matter as long as it’s consistent (remember “ln” is simply log e and is on all scientific calculators). The two best choices (unless the problem says otherwise) is to use log or ln because they are both on your calculator. log b x Remember: All of these rules will be important later on!! Why do we care about logarithms or exponents? Well, let’s talk about some real world applications. Warning: Science ahead! Ever heard of a “decibel” (no pun intended—though it does work out fairly nicely!)? A decibel (abbreviated dB) is a unit of measurement of sound and is based on a logarithmic scale. Near total silence is 0 dB, something 100 times louder than that is 20 dB. A normal whisper is about 15 dB. A normal conversation is about 60 dB. A lawn mower is about 90 dB or 1000 times louder than a normal conversation (for more information on decibels click here). “How is this found?” I seem to hear you ask. Let me show you. If EO is the output voltage of a device and EI is the input voltage then we have the following: E dB gain 20 log O EI Ever wonder how “they” know how old a skeleton that “they” find is? We humans actually have a radioactive element inside us called carbon-14. It’s ok, all living organisms (from trees to people) have it and it really doesn’t do anything bad (we get it from the foods we eat and they get it as a byproduct of photosynthesis). Carbon-14 has a known decay rate (also known as a half-life). Thanks to experimentation, scientists have determined that, with a known quantity of carbon-14, in about 5,700 years half of it will have decayed. Scientists are able to use a technique called Radiocarbon Dating to determine the age of organic fossils (once the organism dies, it no longer gains carbon14—for more information on Radiocarbon Dating click here). The equation that 3 describes this is the following (where A is the amount of radioactive material present at time t and A0 is the amount initially present): A A0 2 t 5700 So, let’s take a look at some examples using what we’ve learned so far. Another fair warning, though: review the following key exponent concepts (it will make things so much easier if you understand these). So, why don’t we work some examples? Sounds like fun, right? Example Solve for x to four decimal places 1. 7 x 12 Answer 7 x 12 log 7 x log 12 take the log of both sides x log 7 log 12 using Rule 7 from above log 12 divide both sides by log 7 to get x alone x log 7 x 1.2770 52 x 1 125 x log 52 x 1 log 125 x Solve for x to four decimal places 2. 52 x 1 125 x 125 53 rewrite 125 as to have a base of 5 and plug it back into the equation 2 x 1 log 5 x log 53 2x 1 log 5 3x log 5 2 x log 5 log 5 3x log 5 after distributing log5 into 2x+1 2 x log 5 3 x log 5 log 5 subtract 3xlog 5 from both sides, bringing it over to the left hand side, then subtract log5 from both sides, bringing it over to the right hand side. x log 5 log 5 after subtracting the terms on the left hand side x log 5 log 5 dividing both sides by –log5 x 1 remember, if the numerator is the same as the denominator, they cancel each other out and leave a 1 in their place 4 Solve for x to four decimal places 3. 3x 3 3x 84 (Hint: 3x 3 3x33 ) 3x 3 3x 3x33 3x 84 84 rewrite after applying the hint notice how both terms on the left x side have a 3 in them? Let’s factor that out. 3x 33 1 84 3x 27 1 84 3x 28 84 84 3x 28 x 3 3 when the bases are the same on both the left hand side and the right hand side (in this case the bases are both 3) then the exponents have to be equal, therefore: x 1 there really was no need to take the log of both sides (though if you did, you will still get the same answer) Solve 4. log( 3x 5) log( 2 x 6) log( 3x 5) log( 2 x 6) 0 0 add log(2x+6) to both sides log( 3x 5) log( 2 x 6) now, put these two equations as exponents to the same base (chose the same base that the log is worked in) 10 log(3 x 5) 10 log(2 x 6) remember 10 is the understood base of a logarithm if there is no subscript attached to the word “log” in the equation 3x 5 2x 6 3x 2x 6 5 from Rule 4 above using basic Algebra concepts (subtracting both 2x and 5 from both sides) x 1 not quite done yet—have to plug this back in to the original equation to make sure we don’t break any logarithm rules (namely the rule about taking the log of a negative number) Another way of looking at this… log( 3x 5) log( 2 x 6) 0 add log(2x+6) to both sides log( 3x 5) log( 2 x 6) if both sides have a log with the same base (remember, this one is base 10 on both sides) we can just set what’s inside the parenthesis to be equal 3x 5 2x 6 3x 2x 6 5 using basic Algebra concepts (subtracting both 2x and 5 from both sides) x 1 as you can see, we still get the same answer Checking… log( 3(1) 5) log( 2(1) 6) 0 log( 8) log( 8) 0 This satisfies the rules of logarithms and, therefore, x 1 is a valid solution of 5 log( 3x 5) log( 2 x 6) Solve 5. log 5 (7 x) log 5 (8 x) log 5 2 2 log 5 (7 log 5 (7 log 5 (7 log 5 (7 log 5 56 56 log 5 log5 5 0 x) log 5 (8 x) log 5 2 2 x) log 5 (8 x) log 5 2 2 x)(8 x) log 5 2 2 from Rule 5 x)(8 x) log 5 2 2 x x 2 56 x x 2 2 x2 x log 5 2 2 after FOIL-ing 2 2 from Rule 6 52 putting both sides as exponents of the same base—choosing 5 as the base because that’s the base of the logarithm x x2 25 multiply both sides by 2 2 56 x x 2 50 subtract 50 from both sides 6 x x 2 0 reorganize to look like a quadratic 56 (putting the squared term first) x 2 x 6 0 factor ( x 3)( x 2) 0 ( x 3) 0 ( x 2) 0 x 3, 2 don’t forget to check to make sure neither solution gives a negative inside the parentheses of any of the logarithms (they don’t) For additional information, check out these websites: -A Review of Logarithms (http://www.sosmath.com/algebra/logs/log1/log1.html) -Logarithmic Equation Solving (http://fym.la.asu.edu/~tturner/MAT_117_online/Logarithms_files/Solving_Log_proble ms/Solving_Log_Problems.htm) -Solving Logarithmic Equations (http://www.sosmath.com/algebra/logs/log4/log47/log47.html) 6