Logarithms and Exponentials

Transcription

1 Logarithms and Exponentials Steven Kaplan Department of Physics and Astronomy, Rutgers University The Basic Idea log b x =? Whoa...that looks scary. What does that mean? I m glad you asked. Let s analyze the equation above. The number b is called the base, x is called the argument, and the? is the exponent. Exponent, what does a logarithm have to do with an exponent. The answer is: logarithms have everything to do with exponents! When you take the logarithm of a number, you are asking the following question: b (the base of the logarithm) raised to what power will give me what I am taking the logarithm of. In math, you can read this as (using the same notation as above): b? = x I.e, when you take the log base b of x, I am asking the question b to what power is going to give me x. The answer to that question is an exponent: the power that b is raised to in order to get x. Let s see an example: log 3 (9) =? Again, our aim here is to figure out what the question mark is. Let s think through it using the idea of what a logarithm is. The question this example is asking is 3 raised to what power is going to give me 9? The answer here is 2 (3 2 = 9). We just did our first logarithm! Let s see another: log 2 (8) =? Ok, same idea here. I want to know what power to which I need to raise two in order to get 8. The answer is 3. (2 3 = 8). How about one more? log 10 (0.001) =? 10 raised to what power is going to give me 0.001? Well, if you know how to use scientific notation, you know how to do this problem! 10 3 will give you 0.001, so 3 is the answer. 1

2 The Commonly Used Bases, e and 10 In your introductory physics courses, you will mainly see logarithms with either a base of e or a base of 10 (logarithms with base e are much more frequent). Both of these have somewhat different notation than what we have seen hitherto: ln(x), The Natural Logarithm These will be what you will see the most. Ln stands for natural logarithm. Why are the letters switched you ask? Probably just to have l in front to be consistent with the other logarithms. ln x is the same thing as log e x. When you encounter natural logs, you will never see them as log e, you will see them as ln. This is to say: log e x = ln x The notation is different, but the meaning is exactly the same. You will never see the former notation, but that is the meaning of ln x: the logarithm of x in base e. This is to say, using what we know about logarithms, the statement ln x is asking e to what power is going to give me x?. A (hopefully) obvious example: ln e 4 =? e to what power is going to give me e 4? Um...well ? Yup! Another (even more) obvious example that will prove very fruitful later on when we get to more complex examples: ln e =? e to what power is going to give me e? Well, I know that something raised to the power of 1 gives you that something back, so that must mean that e 1 = e. Therefore, the answer is 1. This unlocks a very powerful identity for us that can be used with a logarithm of any base b: log b b = 1 b to what power gives me b? Like we reasoned above, 1. log(x), A Logarithm With Base 10 This one you will only see in one or two concepts in physics (it could be that you will never encounter it). For example, the decibel scale of sound intensity is a log 10 scale. Its meaning is the same as any other logarithm: log 10 (x) is an expression symbolizing the question 10 to what power is going give me x? When log 10 is used, you will never see the 10 there. If you just see log, it is most likely base 10. Note: While the vast majority of texts and professors use the ln notation for 2

3 natural logarithms, some will just write log without a base to indicate a natural log. Personally, I think this is terrible notation because they likely also do this for base 10 logarithms. If I have a pet peeve, it is bad notation. Why make things more complicated than they already are? For the sake of completeness, another example: log 1000 =? 10 to what power will give me 1000? Again, for those who know scientific notation, this is simple = 1000, so log 1000 = 3. Logarithmic Identities Ok, every example presented so far has been something you can do in your head. I presented it this way so that you would understand what a logarithm actually is before we get into the harder examples (i.e. the types of things you will actually see in your physics course). Let s take a look at the following example: ln ( ) =? This is one you definitely can t do in your head (unless you ve seen the answer already, but that doesn t count)! For these, you will need a calculator. The answer here is 1 (e 1 = ). How about this one? log 7 (52) On your calculator, you probably have a log key and you probably have an ln key. There s probably no calculator that has a log 7 key. Can you still do this problem on your calculator? Absolutely. You have to use the following identity to transform the example into something your calculator can do. log b (x) = ln (x) ln (b) (1) So, for our case: log 7 (52) = ln (52) ln (7) = Let s check our answer. If the answer is correct, should be very close to (as I rounded the final answer) 52. Indeed, As you can see, we took a logarithm that we couldn t directly use the calculator for, and transformed it into something we could use the calculator for. Another example: log 2 (π) 3

4 . Using the same method as the previous example: log 2 (π) = ln (π) ln (2) Checking our work, Close enough. The less you round, the closer you ll get. Let s go back to the second to last example we had in the natural logs section: ln (e 4 ) =? If the solution wasn t obvious to you, there is a way to simplify the expression. First, note that the following is true for any base b: Therefore: log b (x y ) = y log b (x) (2) ln (e 4 ) = 4 ln (e) = 4 We got the same answer for ln (e 4 ) by doing it both in our heads and by simplifying the expression. Also, note that we used the fact that ln (e) = 1 (Remember, e 1 = e; therefore, ln (e) = 1). Let s take a look at a couple of more examples of logarithms in which we can use this identity to help simplify the expression: log ( ) = 2.7 log (10) = 2.7 Note: for the same reason that ln (e) = 1, log (10) = 1 (i.e = 10) Two more properties of logarithms that will come up are the following: log b (xy) = log b (x) + log b (y) (3) log b ( x y ) = log b (x) log b (y) (4) Problem Solving Techniques Let s look at a few examples and try to simplify them as much as we can: ln (3x) = ln (3) + ln (x) ln (x) ln (3e x ) = ln (3) + ln (e x ) = ln (3) + x ln (e) x ln ( xy e z ) = ln (xy) ln (ez ) = ln (xy) z ln (e) = ln (x) + ln (y) z Of great importance is the concept I have used twice in the examples above: ln (e x ) = x (5) 4

5 Can you tell why this is the case? If not: ln (e x ) = x ln (e) = x We will see why this is so important momentarily. Solving equations: so far, we have just been dealing with a logarithmic expression by itself. We have learned what a logarithm is, various identities they follow, how to evaluate logarithmic expressions, and how to simplify them. Now, we are going to see how they are really used in the context of physics: in equations. First, let us take a look at the following equation: e x = 2 How would you solve for x? The way to do it is this: whenever you have e x = whatever as an equation, you solve for x by taking the natural log of both sides. Watch: e x = whatever ln e x = ln whatever x ln e = ln whatever x = ln whatever Therefore, in the equation e x = 2, x must equal ln 2. This shows another technique that can be employed with equations evolving exponentials. Let s take a look at what we gleaned from our example. More generally, we can see that e ln 2 = 2 e ln x = x (6) This should make sense. We know from the beginning of the chapter that ln x, by definition, is the exponent to which e must be raised to get x. So, e raised to that power should, in fact, give me x! This is exactly what we are seeing. Now, when would we use this? Say that we have an equation where one side is the natural log of an expression involving x (or any variable), and I want to solve for x. If we raise e to both sides of the equation, the natural log term will drop out. Let s see an example. ln (x + 3) = 7 e ln (x+3) = e 7 x + 3 = e 7 x = e

6 One more example for good measure: ln (x 2 ) = 1 e ln(x2) = e 1 x 2 = e x = ± e 6

7 Practice Problems Compute the following logarithms. For the first few, try to do them in your head. For each, write the logarithm out in the form we discussed: if log b (x) = y, then x = b y. Use this to check your answer. This will serve to reinforce the concept of what a logarithm is. 1. log(100) 2. ln(e 3 ) 3. log 3 (27) 4. log 2 (16) 5. log 6 (234) 6. log 1 1(335.6) Solve the following equations for the variable x: 1. e x = x = x2 +10x 5 = 0 4. ln(x 2 ) = 0 5. ln(x 2 + 3x 9) = x2 4 =