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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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Logarithmic Functions

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The logarithic base ten form represents the exponent 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 are looking at the power that we can raise 10 to and the answer that results. So 10 to the 1 is just 10 times 1 which gives us 10. 10 to the 4 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 over 10. Right? Because what it's really saying is 10 inverse, so that's 1 over 10, so that gives us 0.10. And then 10 to the 0, any number to the zeroth power, equals 1. How does this relate to my log function? Well here we have log(ten) gives us 1. Really what this is saying, it's saying that 10 to what number gives me 1? The answer would be 1. Here, log of 10,000. Alright, so think of it like this, we have log of 10,000 so that is really log of 10 to the 4. Whatever the number is here, because remember this is log base 10, what happens here is that this cancel out with this so we get 4 left at the end. Here when we get log of 0.10. What does that really mean? That really means log of 10 to the negative one. So this cancels out with this and gives us negative one. Then we have log of 1. Right? So you'd say here, so log base 10 and then 10 to the one here, well, well, 10 to the 0 here actually sorry, equals 0. So that is what is really going on here. And if we have log(ten) think that as log base ten and then 10 to the 1 equals 1. So we're just converting each of these values, by 10 to some power. The log portion cancels out the 10 and leaves behind the exponent as my final answer. That's how we can c log. I know you guys have calculators, but there may come a point where you have this within a math class or within an MCAT or PCAT or OAT or DAT 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 base 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 1. So here we are going to use this without a calculator. Try to do it without a calculator, see what answer you get, and then afterwards come back and use matches up. I'll help you guys a little bit for the first one. So here, this log is getting distributed to the 1 and to the 10 to the negative 7. Now when that happens, what that means is we have log of 1, and because they're multiplying it really means that we're adding, So it's plus log of 10 to the negative 7. 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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Logarithmic Functions

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So continue where we left off. It says, without using a calculator, determine the answer to the following questions. So remember we have log of 1.0 times 10 to the negative 7 here. Because they are multiplying, we can distribute the log to the 1 and to the 10 to the negative 7. Because they are multiplying, it really means that they are being added together. From up above we saw that log of 1 is equal to 0. And then we saw up above that this is log base 10, so that 10 and this 10 can be thought of canceling each other out. So then what would be left behind would be this, negative 7. So we have 0+-seven, so our answer at the end would be negative 7. For this one, log(1,000). Think about it. How else could I write a1000? I could write it as 10 to the 3. So this cancels out, cancels out. So we would be left with a value of 3. Double check, plug it into your calculator, see if you get the same exact answer. Remember, if you have a number to a power, the times 10 to a power, you should put it in parenthesis in your calculator. Otherwise, you may get the incorrect answer. So you do log, open parenthesis, 1.0 times 10 to the negative 7, close parenthesis, and get your answer of negative 7. So we have done these 2. 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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Logarithmic Functions

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So without using a calculator, determine the answer to the following questions. So here we have log of 1.0 times 10 to the 5. So just like up above the log is getting distributed to the 1 and to the 10 to the 5th. So we're gonna say here because they're multiplying what this really means is we have log(one) +log(105). Remember up above we said that log1 is equal to 0. And then this is log base 10, so this cancels out with this, giving us 5 at the end. So here log of that number would be 5. Now this is easy to remember in terms of figuring out the log of an answer really quickly. But if this number happens to be a number different from 1, then it's best to input 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 log of 1.0 times 10 to any power. And then finally this one here we have log of 0.0001. We want to change this. We are going to say that if we move this over and change to scientific notation, then we have log of 1.0 times 10 to the negative 4. So it just becomes like the one we just did. The log gets distributed to both, so this is log of 1 plus log(10) to the negative 4. Log(one) is just 0. This is base 10 which cancels out with this, which gives me negative 4. So it would be 0 +-four. So the answer here would be negative 4 as 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 pOH of 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 to these log functions here.

Inverse Logarithmic Functions

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

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Inverse Logarithmic Functions

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The inverse or antilogarithmic function can be seen as the opposite of your logarithmic function. We are going to say here, let's take for example, log of x gives me y. If we take the antilog or the inverse log of y 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 log(10) is equal to 2 because remember, log base 10, 100 is really 10 to the 2. The log base 10 of the 10 will cancel out and then this exponent becomes my answer of 2. When we do the inverse or the antilog, what that really means is that we have 10 to the 2, which is equal to 100. So when we take the antilog of a number, what happens is that your x becomes your answer based on the original log function. Now, when will we come into basically the presence of having to use the antilog function? If we take a look here, only when we're dealing with situations, dealing with buffers that's when we might see the use of the antilog function. So for this example here, they are telling us that the Henderson Hasselbalch 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. Our blood is based on the concept of a buffer. A buffer prevents the pH of a solution, in this case our blood, from becoming too acidic or too basic, very quickly. So if you ate something that was acidic and you didn't have buffers in your blood, then your blood will become very acidic very quickly, and you become acidotic. 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 Hasselbalch equation, and it basically gives us a ratio of base to acid within a solution, and specifically our conjugate base to our weak acid. Now using this formula, we are going to see how the antilog function plays a role in finding the answer that we need. We need to figure out the ratio, they tell us, of conjugate base to weak acid. They tell us that pH is equal to 4.17. They tell us that pKa is equal to 3.83. We're gonna plug those numbers in. So pH equals 4 point 17. My pKa is 3.83 plus log of my conjugate base over my weak acid, which I'll abbreviate as cb over w a. We're gonna subtract 3.83 from both sides. So it's gonna give me 0.34 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'll cancel out this and then the anti log will raise this to a power. So 10 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 option c once we round. So what this ratio is telling me, and remember ratio means we'd have two numbers, this number would be over 1. So what this answer is telling me, it's telling me for every one weak acid component, I have 2.18776 of my conjugate base. That'd be the ratio in terms of 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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Natural Logarithmic Functions

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So the natural logarithm, sometimes shortened to just ln, so ln is the natural logarithm, of a value is the exponent to which e must be raised to determine that number. So for example, ln of a 1,000 equals 6.908. E is the inverse of the natural log, so we do e to 6.908, that would give me 1,000. K. So here, the inverse of the natural logarithm again represent, symbolized, by the variable e can be seen as the opposite of the natural logarithm n. So ln=y, so the inverse of my natural log y equals e to the y equals x. Alright so applying this logic, the example below would go as follows. So we have ln(2) equals point 693. So the inverse of my ln, inverse of my natural log which is e, to the point 693 equals 2. Now when do we have to deal with this natural logarithm? We're gonna say that this is very common when we deal with chemical kinetics. So when 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 logarithmic functions, solve for the missing variable in the following question. So they give us ln of x equals negative 2.13 times 10 to the negative one times 12.3 plus ln of 1.25. What we are going to do here is we are going to bring down ln of x. We're going to multiply these together. Don't forget the negative sign that's there. So when we multiply those 2 together, we get negative 2.6199. Then we're gonna add ln of 1.25 to that. So punch in your calculator ln of 1.25. In your calculator you might have to hit second function, look for ln, and then parenthesis 1.25 close parenthesis. When you do that you will get 0.223144. And then when we add those together we get -2.39676 which is still equal to ln. We need to figure out what x is, to do that we have to take the inverse of my natural logarithm. So here, when we take the inverse that will get rid of my ln and then its gonna be e to this value. So whatever the value is, it gets raised to a power. So in your calculator, you may see this button, e to the x. Usually you have to hit second function on your calculator then hit that button, then parenthesis, plug in negative 2.39676 close parenthesis. If you do it correctly, what you are going to get is 0.09 1013. That would be our value here. Here we are not going to 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 logarithm if we have lnx equals some value. To find the x, we have to get rid of the ln, and we do that by taking the inverse of both sides. Taking the inverse gets rid of the ln, and then it raises this number to 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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Logarithmic Relationships

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Hey guys. So here it is going to become important that you guys remember the different operations that occur when we are dealing with logarithmic functions and natural logarithmic functions. So for example, lets take a look at multiplication. So here we have a log of a times b. Now that log is getting distributed to my a and my b. And because they are multiplying with each other, what that translates to is, log+log. And the same thing can be seen with our natural log. So ln a times b, ln gets distributed to both. Because they are multiplying it really means ln of a plus ln of b. Next, we are dealing with division. So when we are 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 b. So when you are multiplying it really means that you are adding each one but when you are dividing and subtracting. Same thing with my LN. So LN of of a minus ln 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 log of a. And same thing for ln. X comes up front so it becomes x equals ln 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 pH and pOH of acids and bases, that's when we have to take into account log functions. And some of the questions require these types of manipulations. Then if we are taking it to the nth root, so here we have log of x, square root, sine, and and a inside. So remember that could be 2, that could be 3, that could be 4. So second root which is square root, third root, 4th root. So remember when we are doing a nth root, what does that really mean? First of all, what that means is it is log of a to the one over x. Okay? So when we change that into a power, it becomes a reciprocal power, so it becomes one over x. And again, like we said before, the exponent can come up front. So this equals 1 over x times log of a. And then here, same thing with ln. It becomes ln of a to the one over x. Again, this can come out front. So this equals one over x times ln of a. So these are the different manipulations that you need to be aware of that will happen when we get to, again, chemical kinetics and when we are dealing with log functions of acids and bases. Based on what we have seen here I say solve the following without the use of a calculator. If log of 3 is equal to approximately 0.48 and log of 2 is equal to approximately 0.30, what will be the value of log of 12? Alright so we have to think of which of these operations we can use. So we wanna find the log of 12. Now an easy way to, to approach this is we could think of what could I do with 32 to give me a value of 12? That's why I gave you those numbers. And the answer is, if I do 3 times 2, that gives me 6. And then if I multiply that by 2 again, that gives me 12. So we can say that log of 12 can be seen as log of 3 times 2 times 2. And we just saw up above when you have things multiplying together, the law gets distributed to all of them. And because there is multiplication under being done, that really means it's log(3) +log(2) +log(2). And so all we're gonna to do now is they tell us what these values are, this is 0.48, this is 0.30, and this is 0.30. So when we plug all that in, that gives me 1.08 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 are dealing with log functions or natural log functions.