Power and Root Functions - Online Tutor, Practice Problems & Exam Prep

Understanding logarithmic functions is essential for comprehending various scientific and mathematical concepts. A logarithm, specifically a base-10 logarithm, determines the power to which the base (10) must be raised to obtain a certain number. For example, log(10) equals 1 because 10 to the power of 1 is 10. Similarly, log(10000) equals 4 since 10 raised to the power of 4 is 10000. The negative exponent, such as log(0.10), translates to log(10^-1), resulting in -1. Moreover, any number raised to the power of zero is 1, making log(1) always equal to 0.

The antilog, or inverse logarithmic function, reverses this process. For instance, if log(x) = y, then 10^y = x. This relationship is pivotal in calculating pH levels using the Henderson-Hasselbalch equation, where the antilog helps determine the ratio of conjugate base to weak acid in a buffer solution.

Natural logarithms (ln), which use the constant e as the base, operate under similar principles. The natural log of a number is the power to which e must be raised to yield that number. For example:

Logarithmic Functions

The logarithmic 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're looking at the powers that we can raise 10 to, and the answer that results. So 10 to the one is just 10 * 1, which gives us 10.

10 1 = 10

10 4 = 10000

10 - 1 = 1 10 = 0.10

And then 10 to the zero. Any number to the 0TH power equals one.

10 0 = 1

How does this relate to my log function? Well here we have log of 10 gives us 1. Really what this is saying? It's saying that 10 to what number gives me one. The answer would be 1 here. Log of 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 log base 10. What happens here is that this will cancel out with this. So it would get 4 left at the end here.

log ( 10 ) = 1

log ( 10000 ) = log ( 10 4 ) = 4

When we get log of 0.10, what does that really mean? That really means log of 10 to the -1. So this cancels out with this and gives us -1. Then we have log of one, right? So you'd say here, so log base 10 and then 10 to the one here. Well, we'll tend to 0 here actually. Sorry, equals 0. So that's what's really going on here.

log ( 0.10 ) = log ( 10 - 1 ) = - 1

log ( 1 ) = 0

And if we have log of 10, think that of as log base 10 and then 10 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 leaves behind the exponent as my final answer. That's how we can see log.

log ( 10 ) = 1

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 one. So here we're going to use this without a calculator. Try to do it without a calculator, see what answer you get and then afterwards come back use a calculator and see if your answer matches up. I hope you guys a little bit for the first one.

So here this log is getting distributed to the one and to the 10 to the -7. Now when that happens, what that means is we have log of one, and because they're multiplying, it really means that we're adding so it's plus log of 10 to the -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.

log ( 1 ) + log ( 10 - 7 )

Logarithmic Functions

So continue where we left off. It says without using a calculator, determine the answer to the following questions. So remember we have a log of 1.0×10-7 here. Because they're multiplying, we can distribute the log to the one and to the 10-7. Because they're multiplying, it really means that they're being added together.

From up above we saw that log of one is equal to 0 and then we saw up above that this is log base 10 so that 10 in this 10 can be thought of canceling each other out. So then what would be left behind would be this -7. So we have 0 + -7. So our answer at the end will be -7 for this one.

Log of 1000. Think about it, how else could I write 1000? I could write it as 103, so this cancels out. Cancels out. So we'd 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 parentheses in your calculator. Otherwise you may get the incorrect answer. So you do log ( 1.0×10-7 ) and get your answer of -7.

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.

Logarithmic Functions

So without using a calculator, determine the answer to the following questions. So here we have a log of 1.0 * 10 to the five. So just like up above, the log is getting distributed to the one and to the 10 to the 5th. So we're going to say here because they're multiplying. What this really means is we have log of one plus log of 10 to the 5th. Remember up above we said that log of one is equal to 0 and then this is log base 10. So this cancels out with this giving us five 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 1:00, 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 * 10 to any power.

And then finally this one. Here we have log of 0.0001. We want to change this, so we're going to say that if we move this over and change it to scientific notation, then we have log of 1.0 * 10 to the -4. So just becomes like the one we just did. The law gets distributed to both, so this is log of one plus log of 10 to the -4. Log of one is just zero. This is base 10, which cancels out with this, which gives me -4. So it would be 0 + -4. So the answer here would be -4 as my final answer.

So these are just the basics when it comes to log functions. They're going to 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 blog functions here.

Inverse Logarithmic Functions

The inverse or anti logarithmic function can be seen as the opposite of your logarithmic function. We're going to say here. Let's say for example log⁢X gives me Y. If we take the antilog or the inverse log of Y then that would give me 10Y 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 two because remember log⁢10100 is really 102. The log⁢10 and the 10 will cancel out and then this exponent becomes my answer of two when we do the inverse or the antilog. What that really means is that we have 102 which is equal to 100.

OK, 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 into basically the presence of having to use the antalog 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 antilog function. So for this example here, they're telling us that the Henderson Hasselbeck 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 and 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 going to say with buffers, we have the Henderson Hasselbeck 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're 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 the pKa is equal to 3.83. We're going to plug those numbers in so pH equals 4.17. My pKa is 3.83 plus log of my conjugate base over my weak acid, which I'll abbreviate as CB over Washington. We're going to subtract 3.83 from both sides, so it's going to 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'll cancel out this and then the anti log will raise this to a power. So 10.34 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 their 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 one. So what this answer is telling me, it's telling me for everyone weak acid component I have 2.18776 of my conjugate base that be the ratio in terms of the distribution between my conjugate base and my weak acid.

Natural Logarithmic Functions

So the natural logarithm sometimes shortened to just lane. So lane is the natural logarithm of a value is the exponent to which E must be raised to determine that number. So for example, ln⁡(1000)=6.908, E is the inverse of the natural log, so we do e6.908. That would give me 1000.

OK, so here the inverse of the natural logarithm again represents, symbolized by. The variable E can be seen as the opposite of the natural logarithm a. So lane of X = y. So the inverse of my natural log y = ey = X. All right, so let's so applying this logic, the example below would go as follows. So we have ln⁡(2)=0.693, so the inverse of my lane inverse of my natural log, which is e0.693 = 2.

Now when do we have to deal with this natural logarithm? We're going to 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 lane 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⁡(X)=-2.13⁢×10⁻1⁢×12.3+ln⁡(1.25). So what we're going to do here is we're going to bring down Allen of X. We're going to multiply these together. Don't forget the negative sign that's there. So when we multiply those two together, we get -2.6199. Then we're going to add lane of 1.25 to that.

So punch in your calculator, lane of 1.25. In your calculator you might have to hit second function, look for lane and then ( 1.25 ). When you do that, you'll get .223144 and then when we add those together we get -2.39676 which is still equal to lane of X. 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'll get rid of my lane and then it's going to be e-2.39676. So whatever the value is it, 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 -2.39676 ). If you do it correctly, what you're going to get is .091013. That would be our value here.

Here we're 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 ln⁡(X)=some value, to find the X we have to get rid of the lane, and we do that by taking the inverse of both sides. Taking the inverse gets rid of the lane, and then it raises this number to a power. It becomes ethat number. That'll help us find our missing variable.

Logarithmic Relationships

Hey guys, so here it's going to become important that you guys remember the different operations that occur when we're dealing with logarithmic functions and natural logarithmic functions. So for example, let's take a look at multiplication. So here we have a log of a * b. Now that log is getting distributed to my A and my B, and because they're multiplying with each other, what that translates to is log of A plus log of B. And the same thing can be seen with our natural log. So ln⁡(A⁢B) gets distributed to both. Because they're multiplying, it really means ln⁡(A)+ln⁡(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 / B becomes log of A minus log of B. 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⁡(A)-ln⁡(B). What happens when we raise it to a power? Woah, 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⁡(A), X comes up front, so it becomes X ⁡(ln⁡(A)).

So there's a little bit of trick functions when it comes to these types of situations. This is going to 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're taking it to the NTH route. So here we have a log of X square root sign and and a inside. So remember that could be two. That could be three. That could be 4. So second root, which is square root, third root, 4th root.

So remember, when we're doing a NTH root, what does that really mean? First of all, what that means? It's log of A to the 1 / X. OK, so when we change that into a, power becomes the reciprocal power, so it becomes 1 / X. And again like we said before, the exponent can come up front, so this equals 1 / X times log of a. And then here, same thing with ln⁡(A), it becomes ln⁡(A)^(1/X). Again, this can come out front, so this equals 1 / X times ln⁡(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're dealing with log functions of acids and bases.

Based on what we've seen here, I say solve the following without the use of a calculator. If log of three is equal to approximately 48 and log of two is equal to approximately 30, what will be the value of log of 12? All right, so we have to think of which of these operations we can use. So we want to find log of 12. Now an easy way to to approach this is we can think of what could I do with three and two to give me a value of 12. That's why I gave you those numbers. And the answer is if I do 3 * 2 that gives me 6 and then after I multiply that by two again that gives me 12.

So we can say that log of 12 can be seen as log of 3 * 2 * 2 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 on being done, that really means it's log of three plus log of two plus log of two. And so all we're going to do now is tell us what these values are. This is 0.48, this is 0.30, and this is .30 O. When we plug all that in, that gives me 1.08 as my final answer, O. 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.