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5. BONUS: Mathematical Operations and Functions

**Logarithmic** and **natural logarithmic functions** of numbers.

Logarithmic Functions

The logarithmic base 10 form represents the exponent that 10 must be raised in order to obtain that specific number.

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the longer thick base 10 form represents the exponents that 10 must be raised in order to obtain that specific number. Now what does this really mean? Let's take a look at these examples here. So here we're looking at the power that we can raise 10 to and the answer that results. So 10 to the one is just 10 times one, which gives us 10. 10 of the four is just 10 times 10 times 10 times 10 which gives US 10,000 10 to the negative. One is equivalent to just saying 1/ right, because what it's really saying is 10 inverse. So that's 1/10. So that gives us 100.10 and then tend to the zero. Any number to the zeroth power equals one. How does this relate to my log function? Well, here we have long of 10 gives us one really what this is saying. It's saying that 10 toe what number gives me? Give me one. The answer would be one. Here. Log off. 10,000. All right. So think of it like this. We have log of 10,000. So that's really log of 10 to the four whatever the number is here. Because, remember, this is long based 10. What happens here is that this will cancel out with this. So we get four left at the end here when we get log off 0.10. What does that really mean? That really means log off 10 to the negative one. So this council's out with this and gives us negative one. Then we have log of one. Right? So you'd say here. So log based 10 and then tend to the one here. Oh, will tend to zero. You're actually sorry. Equals zero. So that's what's really going on here. And if we have log off 10 think that as log based 10 and then tend to the one equals one. So we're just converting each of these values by 10 to some power. The log portion cancels out the 10 and lose behind the exponents as my final answer. That's how we can see log. I know you guys have calculators, but there may come a point where you have this within the math class or within an M cat or P cat or O A. T or D 80. Later on much later on after you've taken all these science courses where you have to understand these relationships. And this is how our log based function is connected to multiples of 10. Knowing that helps us to get to the answer. Now, understanding this, try to see if you can answer example one. So here we're gonna use this without a calculator. Try to do without a calculator to see what answer you get, and then afterwards come back, use a calculator and see if your answer Masters up matches up. I hope you guys a little bit for the first one. So here this law is getting distributed to the one end to the 10th of negative seven. Now, when that happens, what that means is we have long of one. And because they're multiplying, it really means that we're adding. So it's plus log of 10 to the negative seven. See if you guys can figure out what the answer is without using a calculator, come back and see if your answer matches up with mine.

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so continue where we left off. It says, without using a calculator, determined the answer to the following questions. So remember, we have a lot of 1.0 times 10 to the negative seven here because they're multiplying. We can distribute the log till the one and two. The 10th negative seven. Because they're multiplying really means that they're being added together from up above. We saw that log of one is equal to zero. And then we saw up above that. This is Law based 10. So that 10 in this tin can be thought of canceling each other out. So then what will be left behind would be this negative seven. So we have zero plus negative. Seventh or answer at the end will be negative. Seven for this one. Log off 1000. Think about it. How else could I write 1000? I could write it as 10 to the three. So this cancels out, cancels out. So we'd be left with a value of three. Double check plug into your calculator. See if you get the same exact answer. Remember, if you have a number, tow a power. The times 10 to the power you should put it in parentheses in your calculator. Otherwise, you may get the incorrect answer. So you do log open parentheses 1.0 times 10 to the negative seven close parentheses and get your answer of negative seven. So we've done these two attempt to do this next example on your own. See if you get the answer. You can double check by punching it into your calculator or by coming back and taking a look at my video explanation on how to approach this question as well.

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so, without using a calculator, determine the answer to the following questions. So here we have long of 1.0 times 10 to the five. So just like up above, the log is getting distributed to the one end to the 10 to the fifth. So we're gonna say here because they're multiplying. What this really means is we have long of one plus log off 10 to the fifth. Remember up above, we said that log of one is equal to zero, and then this is logged based 10. So this cancels out with this, giving us five at the end. So here, log of that number would be five. Now, this is easy to remember in terms of figuring out the log of an answer really quickly. But if this numbers happens to be a number different from one, then it's best toe and put it into your calculator. Because if it's log of any other number, this part won't be equal to 10 and you haven't memorized what those numbers are because you have your calculator. So again, this works best when we have long at 10 times 10 to any power, and then finally this one here we have a log of 10. We want to change this. So we're going to say that if we move this over and change to scientific notation, then we have log off 10 times 10 to the negative four. So it just becomes like the one we just did. The law gets distributed to both. So this is log of one plus log off. 10 to the negative. Four long of one is just zero. This is base 10 which cancels out with this, which gives me negative four. So it's zero plus negative four. So the answer he would be negative for is my final answer. So these are just the basics. When it comes to log functions, they're gonna become important when we're talking about chapters dealing with chemical kinetics also will come in handy when we're talking about determining Ph or P o. H off different solutions. That's when log functions really come out. And we have to do different types of mathematical manipulations to get our answer at the end. But for now, just remember the basics when it comes toe these blogged functions here

Inverse Logarithmic Functions

The inverse or anti-logarithmic function is the opposite of the logarithmic function.

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the inverse or anti log horrific function can be seen as the opposite of your logo rhythmic function. We're gonna say. Here, let's take for example, Log of X gives me why. If we take the anti log or the inverse log off, why, then that would give me 10 to the Y, which is equal to X. So what does that mean? If we take a look, let's apply to this example. Here we have a log of 10 is equal to two because remember, Long based 10 100 is really tend to the to the log based 10 of the 10 will cancel out, and then this expo have become my answer of to when we do the inverse or the anti log. What that really means is that we have 10 to the to, which is equal to 100. Okay, so when we take the anti log off a number, what happens is that your ex becomes your answer based on the original log function. Now, when we come in tow into basically the presence of having to use the antilock function if we take a look here normally, when we're dealing with situations dealing with buffers. That's when we might see the use of the anti log function. So for this example here, they're telling us that the Henderson Hasselbach equation is a useful equation for the determination of the ph of a buffer. Now, buffers are incredibly important in chemistry as well as in biology. Are blood is based on the concept of a buffer? Ah, buffer prevents the pH of a solution in this case, our blood from becoming too acidic or two basic very quickly. So if you ate something that was acidic and you didn't have buffers in your blood and your blood will become very acidic very quickly and you become acid attic. If you ate something that was very basic, then you would have basic blood, which again would cause harm to your organs. That's why our blood has within it carbonic acid and sodium bicarbonate to balance one another out. We're gonna say with buffers we have the Henderson has a back equation, and it basically gives us a ratio off base toe acid within a solution and specifically our conjugate base toe are weak acid. Now, using this formula, we're gonna see how the anti log function plays a role in finding the answer that we need. We need to figure out the ratio they tell us of conjugal based a weak acid. They tell us the P H is equal to 417 They tell us the P K is equal to 383 We're gonna plug those numbers in, so Ph equals 4.17 My Pekka is 3.83 plus log of my conjugate base over my weak acid, which I'll abbreviate SCB over W A. We're going to subtract 33 from both sides, so it's gonna give me 34 equals log of conjugate base over weak acid. So here we just want to find the ratio by itself. So we need to take the anti log in order to get rid of this log function here. So when I take the anti log of both sides that will cancel out this and then the anti log will raise this toe a power so tend to the 0.34 power gives me the ratio of conjugate base over weak acid. When we punch that into our calculators, we get 2.18776 as our answer for the ratio. So the closest answer to this would be options, see, once we round. So what this ratio was telling me and remember, ratio means we'd have two numbers. This number would be over one. So what this answer is telling me? It's telling me for everyone weak acid component. I have 2.18776 off my conjugate base that be the ratio in terms off the distribution between my conjugate base and my weak acid.

Natural Logarithmic Functions

The **natural logarithmic function ln **is the exponent to which e must be raised to determine that number.

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so the natural logarithms sometimes shortened to just Ln so Alan is the natural longer rhythm off a value is the exponents which e must be raised to determine that number. So, for example, Ellen of 1000 equals 6.908 e is the inverse of the natural log. So we do e to 6.908 That would give me 1000. Okay, so here, the inverse of the natural logarithms again represent, symbolized by the variable e can be seen as the opposite of the natural logarithms. And so, Alan of X equals y. So the inverse of my natural law y equals e to the y equals X. All right, so So applying this logic, the example below would go as follows. So we have Ellen of two equals 20.693 So the inverse of my Ln inverse of my natural law, which is e to the 0.693 equals two. Now, when do we have to deal with this natural logarithms? We're going to say that this is very common when we deal with chemical kinetics. So we deal with the chapter dealing with chemical kinetics. You will see Ln being used now, using this logic, we can help to solve this question. Here, here it says, based on your understanding of natural law, go rhythmic functions solve for the missing variable in the following question. So they give us Ellen of acts equals negative. 2.13 times 10 to the negative one times 12. plus Allen of 1 to 5. So we're gonna do here is we're gonna bring down Al Innovex. We're going to multiply these together. Don't forget the negative sign that's there. So we multiply those two together we get negative 26199 Then we're gonna add Allen of 1 to 5 to that. So punch in on your calculator Ln off 1. in your calculated, you might have to hit second function. Look for Ln and then parentheses. 1.25 close parentheses. When you do that, you'll get 0. And then when we add those together we get negative 2.39676 which is still equal toe Elena backs. We need to figure out what X is to do that we have to take the inverse of my natural logarithms. So here, when we take the inverse, that'll get rid of my Ln and then it's gonna be e to this value. So whatever the value is, it gets raised to a power. So when you're calculating, you may see this button e to the X. Usually you have to hit second function on your calculator, then hit that button, then parentheses. Plug in negative. 239676 Close parentheses. If you do it correctly, what you're gonna get is 2396760. That would be our value. Here, here, we're not gonna worry about Sig Figs. Let's just worry about getting the correct answer. So this will be my value for X Again. We use the inverse of the natural law algorithm if we have Ln X equals some value. To find the X, we have to get rid of the l N. And we do that by taking the inverse of both sides taking the inverse gets rid of the L N. And then it raises this number. Tow a power. It becomes e to that number that will help us find our missing variable

Logarithmic Relationships

The similarities between logarithmic and natural logarithmic functions are outlined below.

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Hey, guys. So here is gonna become important that you guys remember the different operations that occur, but we're dealing with log arrhythmic functions and natural algorithmic functions. So, for example, let's take a look. Multiplication. So here we have a log of a times B. Now that log is getting distributed to my A and my B because they're multiplying with each other. What that translates to is log of a plus log of B, and the same thing could be seen with our natural log. So, Ellen, a times B. Allen gets distributed to both because they're multiplying. It really means Allen of a plus, Allen of B. Next, we're dealing with division. So when we're dividing them, what does that really mean? Well, that means here that log of a divided by B becomes log of a minus log of the so when you're multiplying, it really means you're adding each one. But when you're dividing and subtracting same thing with my Ln Ln off a minus. Ellen of B. What happens when we raise it to a power? Well, when we raise it to a power, what you need to realize here is that this power can move up front of my log function. So this becomes X times long of a and same thing for Ln X comes up front. So becomes X equals a line of a. So there's a little bit of trick functions when it comes to these types of situations. This is gonna become important when we're dealing with chapters dealing with chemical kinetics. Because with those chapters, we do have to manipulate problems that contain Ln and then when it comes to P H and P o. H of acids and bases, that's when we have to take into account log functions. And some of the questions required these types of manipulations. Then if we're taking it to the 10th root. So here we have a log of X square root sign on a inside, remember That could be, too. That could be three. That could be four. So second route, which is square root, third root, fourth root. So remember what we're doing. A and Root, What does that really mean? First of all, what that means it's log off a to the one over x. Okay, so when you change that into a power becomes the reciprocal power so becomes one over X And again, Like we said before the expo, you can come up front. So this equals one over X Times log of a and then here's same thing with Ln it becomes Ln off a to the one over X again, this could come out front. So this equals one over x times Allen of a. So these are the different manipulations that you need to be aware off that will happen when we get to again chemical kinetics. And when we're dealing with log functions, off acids and bases based on what we've seen here, I say solve the phone without the use of a calculator. If log of three is equal to approximately 48 and log of two is equal to approximately 30 will be the value of log of 12. All right, so we have to think of which of these operations we can use. We want to find a log of 12 now an easy way, Thio to approach This is we could think of What could I do with three and to to give me a value off 12? That's why I gave you those numbers and the answer is if I do three times to that, gives me six, and then I find Multiply that by two again. That gives me 12 so we can say that Log of 12 can be seen as log off three times to times two, and we just saw up above. When you have things multiplying together, the law gets distributed to all of them. And because there's multiplication under being done, that really means it's log of three, plus log of to plus log of tip and soul. All we're gonna do now is they tell us what these values are. This is 48 this is 30 and this is 30 so we plug all that in. That gives me 1.8 as my final answer. So that would be my log of 12. And we did it without the use of a calculator, because we understand the different relationships that arise when we're dealing with log functions or natural log functions.

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