Introduction to Logarithms



Introduction to LogarithmsIn its simplest form, a logarithm answers the question:How many of?one number?do we multiply to get?another number?Example: How many?2s do we multiply to get?8?Answer:?2 × 2 × 2 = 8, so we needed to multiply?3?of the?2s to get?8So the logarithm is 3How to Write itWe write "the number of 2s we need to multiply to get 8 is 3" as:log2(8) = 3?So these two things are the same:The number we are multiplying is called the "base", so we can say:"the logarithm of 8 with base 2 is 3"or "log base 2 of 8 is 3"or "the base-2 log of 8 is 3"Notice we are dealing with three numbers:the?base: the number we are multiplying (a "2" in the example above)how many times to use it in a multiplication (3 times, which is the?logarithm)The number we want to get (an "8")More ExamplesExample: What is?log5(625)?... ?We are asking "how many 5s need to be multiplied together to get 625?"5 × 5 × 5 × 5 = 625, so we need?4?of the 5sAnswer:?log5(625) = 4Example: What is?log2(64)?... ?We are asking "how many 2s need to be multiplied together to get 64?"2 × 2 × 2 × 2 × 2 × 2 = 64, so we need?6?of the 2sAnswer:?log2(64) = 6Common Logarithms: Base 10Sometimes a logarithm is written?without?a base, like this:log(100)This?usually?means that the base is really?10.It is called a "common logarithm". Engineers love to use it.On a calculator it is the "log" button.It is how many times we need to use 10 in a multiplication, to get our desired number.Example:?log(1000) =?log10(1000) = 3Basic Log Rules / Expanding?????Logarithmic ExpressionsYou have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that?x3?×?x5?equalsx8?because you can add the exponents. There are similar rules for logarithms.Log Rules:1)?logb(mn) =?logb(m) +?logb(n)2)?logb(m/n) =?logb(m) –?logb(n)3)?logb(mn) =?n?·?logb(m)In less formal terms, the log rules might be expressed as:1) Multiplication inside the log can be turned into addition outside the log, and vice versa.2) Division inside the log can be turned into subtraction outside the log, and vice versa.3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.Warning: Just as when you're dealing with exponents, the above rules work?only?if the bases are the same. For instance, the expression "logd(m) +?logb(n)" cannot be simplified, because the bases (the "d" and the "b") are not the same, just as?x2?×?y3?cannot be simplified (because the bases?x?and?y?are not the same).Expanding logarithmsLog rules can be used to simplify expressions, to "expand" expressions, or to solve for values.Expand?log3(2x).When they say to "expand", they mean that they've given you one log expression with lots of stuff inside it, and they want you to use the log rules to take the log apart into lots of separate logs, each with only one thing inside. That is, they've given you one log with a complicated argument, and they want you to convert this to many logs, each with a simple argument.??I have a "2x" inside the log. Since "2x" is multiplication, I can take this expression apart and turn it into an addition outside the log:log3(2x) =?log3(2) +?log3(x)The answer they are looking for is:log3(2) +?log3(x)Do not try to evaluate "log3(2)" in your calculator. While you would be correct in saying that "log3(2)" is just a number, they're actually looking here for the "exact" form of the log, as shown above, and not a decimal approximation from your calculator.Expand?log4(?16/x?).I have division inside the log, which can be split apart as subtraction outside the log, so:log4(?16/x?) =?log4(16) –?log4(x)The first term on the right-hand side of the above equation can be simplified to an exact value, by applying the basic definition of what a?logarithm?is:log4(16) = 2Then the original expression expands fully as:log4(?16/x?) =?2 –?log4(x)Always remember to take the time to check to see if any of the terms in your expansion (such as thelog4(16)?above) can be simplified.Expand?log5(x3).???Copyright ? Elizabeth Stapel 2002-2011 All Rights ReservedThe exponent inside the log can be taken out front as a multiplier:log5(x3) = 3 ·?log5(x) =?3log5(x)Expand the following:?????????????The?5?is divided into the?8x4, so split the numerator and denominator by using subtraction:Don't take the exponent out front yet; it is only on the?x, not the?8, and you can only take the exponent out front if it is "on" everything inside the log. The?8?is multiplied onto the?x4, so split the factors by using addition:log2(8x4) –?log2(5) =?log2(8) +?log2(x4) –?log2(5)The?x?has an exponent (which is now "on" everything inside its log), so move the exponent out front as a multiplier:log2(8) +?log2(x4) –?log2(5)??? =?log2(8) +?4log2(x) –?log2(5)Since?8?is a power of?2, I can simplify the first log to an exact value:??log2(8)?+ 4log2(x) –?log2(5)?? =?3?+ 4log2(x) –?log2(5)Each log contains only one thing, so this is fully simplified. The answer is:3 + 4log2(x) –?log2(5)Expand the following:?????????????Use the log rules, and don't try to do too much in one step:Then the final answer is:???Copyright ? Elizabeth Stapel 2002-2011 All Rights Reservedlog3(4) + 2log3(x?– 5) – 4log3(x) – 3log3(x?– 1)The logs rules work "backwards", so you can simplify ("compress"?) log expressions. When they tell you to "simplify" a log expression, this usually means they will have given you lots of log terms, each containing a simple argument, and they want you to combine everything into one log with a complicated argument. "Simplifying" in this context usually means the opposite of "expanding".Simplify?log2(x) +?log2(y).Since these logs have the same base, the addition outside can be turned into multiplication inside:log2(x)?+?log2(y) =?log2(xy)The answer is?log2(xy).?Copyright ? Elizabeth Stapel 2002-2011 All Rights Reserved??Simplify?log3(4)?–?log3(5).Since these logs have the same base, the subtraction outside can be turned into division inside:log3(4)?–?log3(5) =?log3(4/5)The answer is?log3(4/5).Simplify?2log3(x).The multiplier out front can be taken inside as an exponent:2log3(x) =?log3(x2)Simplify?3log2(x) – 4log2(x?+ 3) +?log2(y).I will get rid of the multipliers by moving them inside as powers:3log2(x) –?4log2(x?+ 3)?+?log2(y)????? =?log2(x3) –?log2((x?+ 3)4)?+?log2(y)Then I'll put the added terms together, and convert the addition to multiplication:log2(x3) –?log2((x?+ 3)4)?+?log2(y)????? =?log2(x3)?+?log2(y)?–?log2((x?+ 3)4)????? =?log2(x3y) –?log2((x?+ 3)4)Then I'll account for the subtracted term by combining it inside with division: ................
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