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

The logarithmic function, particularly base ten, represents the exponent needed to obtain a number from 10. For example, $\log \left(1000\right)=3$ because $10^3=1000$. The antilogarithm reverses this, yielding the original number. Natural logarithms, denoted as $\ln$, use the base $e$. Understanding these concepts is crucial in fields like chemical kinetics and buffer solutions, where equations like the Henderson-Hasselbalch equation apply.

Logarithmic Functions

The logarithmic base ten form represents the exponent that 10 must be raised to 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 101 is just 10 times 1 which gives us 10. 104 is just 10 times 10 times 10 times 10 which gives us 10,000. 10-1 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 100, 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(104). 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(10-1). 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 101 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 calculate 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 see if your answer 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 10-7. 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 10-7. 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.

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 log (1.0 × 10-7). Because they are multiplying, we can distribute the log to the 1 and to the 10-7. Because they are multiplying, it really means that they are being added together. From up above we saw that log (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+(-7), so our answer at the end would be negative 7.

For this one, log (1000). Think about it. How else could I write 1000? I could write it as 103. 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 (1.0 × 10-7), and get your answer of negative 7.

So we have 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. Here we have log(1.0×105). Just like above, the log is distributed to the 1 and to 105. Because they are multiplying, this really means we have log(1) + log(105). Remember, we said that log(1) is equal to 0. And since this is log base 10, this simplifies to 5. So here, log(1.0×105) would be 5. This technique is easy to remember for quickly determining the log of a result. However, if the number is different from 1, it's best to use a calculator since this part won't equal to 10 and you haven't memorized those numbers.

Finally, here we have log(0.0001). We want to convert this into scientific notation, giving us log(1.0×10−4). The log is distributed to both, resulting in log(1) + log(10−4). Since log(1) is just 0, and this being base 10, the overall result is -4. The answer here would be -4 as the final answer.

These are just the basics when it comes to log functions. They are going to become important when discussing chapters dealing with chemical kinetics. They will also come in handy when determining the pH or pOH of different solutions. That's when log functions really come out, and we have to perform various 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 antilogarithmic function can be seen as the opposite of your logarithmic function. We are going to say here, let's take for example, logx 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 log10 is equal to 2 because remember, log base 10, 100 is really 102. 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 102, 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 would 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 4.17. My pKa is 3.83 plus log of my conjugate base over my weak acid, which I'll abbreviate as cb over wa. 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 antilog in order to get rid of this log function here. So when I take the antilog of both sides that'll cancel out this and then the antilog will raise this to a power. So 100.34 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 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

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. For example, ln of 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. So here, the inverse of the natural logarithm, again symbolized by the variable e, can be seen as the opposite of the natural logarithm. 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. We have ln2=0.693. So the inverse of my ln, inverse of my natural log which is e, to 0.693 equals 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 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:

"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. When we multiply those 2 together, we get negative 2.6199. Then we're going to 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 lnx=-2.39676. 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 it's going to 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 approximately 0.091013. That would be our value for x. Again, we use the inverse of the natural logarithm if we have ln x 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. That will help us find our missing variable.

Logarithmic Relationships

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, let's take a look at multiplication. Here we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mo>(</mo><mi>a</mi><mo>×</mo><mi>b</mi><mo>)</mo></math>. 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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mi>a</mi><mo>×</mo><mi>b</mi><mo>)</mo><math> really means <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>ln</mi><mo>(</mo><mi>b</mi><mo>)</mo></math>. Next, we are dealing with division. So when we are dividing them, what does that really mean? Well that means here that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mo>(</mo><mi>a</mi><mo>÷</mo><mi>b</mi><mo>)</mo></math> becomes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>-</mo><mi>log</mi><mo>(</mo><mi>b</mi><mo>)</mo></math>. So when you are multiplying it really means that you are adding each one but when you are dividing you're subtracting. The same thing with my LN. So <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>-</mo><mi>ln</mi><mo>(</mo><mi>b</mi><mo>)</mo></math>. 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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>×8