Introduction to Logarithms - Video Tutorials & Practice Problems

Logarithms Introduction

Hey, everyone, whenever we solve polynomial equations like X cubed is equal to 216 we can simply do the reverse operation in order to isolate X. So here, since X is being a cubed, I could simply take the cube root of both sides in order to cancel that isolating X giving me my answer that X is equal to the cube root of 216. But what of my variable X is instead in my exponent like this? Two to the power of X equals eight. How are we going to isolate X here? Well, I know that looking at this, I can just think, OK, how many times does two need to be multiplied by itself in order to give me eight? And I know that my answer is simply X is equal to three. But what if I'm given something like two to the power of X is equal to 216? I really don't want to have to multiply two enough times in order to get up to that 216. So is there not just an operation I can do in order to cancel out that two and leave me with X. Well, here I'm going to show you that there is an operation that does just that because the reverse or inverse of an exponential is actually taking the logarithm or the log. Now, the first time you work with logs, they can be a little bit overwhelming. But here I'm going to walk you through exactly what a log is and how we can use it to actually make our lives easier, especially when working with exponents. So now that we know that the reverse or inverse operation of an exponential is simply taking the log, we can go ahead and take the log of both sides of this equation in order to isolate X. So I take a log of two to the power of X and that's equal to a log of 200 16. But we actually do need to consider one more thing here because whenever we canceled the three on our X cubed, we took the cube route, we didn't take the square root or the fourth route or anything else, we took the cube route in order to cancel that three. And we need to consider something similar when working with logs because logs and exponentials need to have the same base as each other in order to cancel. So here, since I have an exponential of base two, I want my log to have a base of two as well. So really, I want to take a log base two two to the power of X. And then on the other side, a log base two of 216. Now that my log and my exponential have the same base, then it's going to fully cancel out, leaving me with just X is equal to a log base two of 216. Now it's fine to leave it in this form here. This is actually called our logarithmic form. And we're later going to learn how to fully evaluate these and get a number. But for now, we're just going to keep it in that log form. Now that we're here, what exactly does this statement mean? We have log base of 216 while a log is actually giving us the power that some base must be raised to in order to equal a particular number. But what does all of that mean while looking at our function here or our equation here log base two of 216? This is really saying, OK, what power does two need to be raised to in order to give me 216. Now, this statement here I mentioned is in its logarithmic form and it's actually an equivalent statement to our very first equation two to the power X is equal to 216. This is just in its exponential form and we basically translated it into its logarithmic form. Now, we're going to have to do this for multiple statements, we're going to have to translate and convert expressions between these two forms. So diving a bit deeper in converting between these two forms. Let's start by taking this equation in its exponential form three to the power of X equals 81 and putting it into log form. Now, whenever we convert between these two forms, no matter what we're going to or from, we're always going to at the same place, we're going to start with our base. So this base three of our exponent is going to become the base of our log. So I start here with log base three. Now, once I have my base, I'm going to circle to the other side of my equal sign to that 81. So I have a log base three of 81. And then once I have that 81 I'm going to circle back to the other side of my equal sign. And that's what log base three of 81 is going to equal. So that's equal to X. Now, I have my statement in its log form. Log base three of 81 equals X is equivalent to three to the power of X equals 81. Now that we've seen going from exponential to log form, let's go in the reverse direction from log to exponential form. So starting with this statement here, X is equal to log base four of 64. We're going to start at that same place, we're going to start with our base. So here, my log has a base of four that becomes the base of my exponential. So I have a log base four and I have that four. And now I want to raise four to a power and I'm going to raise it to the power that is on the other side of my equal sign. So start with your base, go to the other side of your equal sign. In this case, I get X here. So four to the power of X and then finally circling back to the other side of our equal sign. That is our final number here, 64. So four to the power of X is equal to 64 is an equivalent statement in exponential form from this X equals a log base four of 64. Now, I know that that might seem alike a lot right now. So let's just walk through some examples together. So let's start with this X is equal to log base five of 800. Since that is in its log form, I have a log right there. I want to go ahead and put this in exponential form. So remember we're going to start at the same place every single time, no matter what we're starting with, we always start with that base. So here I have log base five. So I'm going to start with an exponential of base five. Now once I have that base, I'm going to circle to the other side of my equal sign and get that X so five to the power of X and then I'm going to circle back to the other side of my equal sign and this is equal to 800. So start with your base other side of your equal sign circle back to where you started. So five to the power of X equals 800 is my equivalent statement in its exponential form. Let's look at another example here we have log base two of 16 equals four. Now where are we going to start here? Well, we want to start with our base, of course. So I have this log base two. So that becomes the base of my exponent. So I have two and then I go to the other side of my equal side two to the power of four and then a circle back to where you started two to the power of four equals 16. Now this is great because we can actually see that this is a true statement two to the power of four is equal to 16. So I know that I wrote that correctly. Let's look at one final example here, here we have 10 to the power of X 4500. So this is in its exponential form, we want to go ahead and put this in log form. So, so remember we're going to start with our base. So here, my exponent has a base of 10 and this becomes the base of my log. So I start with log base 10 and then I circle to the other side of my equal sign and get 4500. So log base 10 of 4500 and that is equal to circling back to the side that I started on log based 10 of 4500 is equal to X. And that is my equivalent statement. But looking at this log based 10 of 4500, this log based 10 is actually a sort of special type of log. So log based 10 is actually known as the common log because it is occur, it occurs so frequently. So it can actually just be written as a log. It gets its own special notation. It's just log because it is that common. So I could really just write this as a log of 4500 and that is equal to X. Now, this also has its own button on your calculator. If you need to evaluate which we'll do in the future, it's just the log button. So now that we know a little bit more about logs, and we've even seen our first common log. Let's go ahead and get some more practice.

The Natural Log

Hey, everyone, we just learned about a frequently occurring log log base 10. And there's actually another log that occurs rather frequently log base E called the natural log. Now, don't worry, we're not going to have to learn how to do anything new here. Here. I'm going to walk you through how we can treat log base E just as we would, any other log of any other base, just with a special notation. So let's go ahead and get into it. Now, log base 10, our common log, we would write as log and similarly log base E gets its own notation. It's always written as Ln. Now a way to remember that is because this is the natural log. If we take the first letter of each of those words, natural log and we reverse them, we get Ln, which we're always going to read out as natural log. Now, the natural log also has its own special button on your calculator. The Ln button for whenever you need to use that to evaluate something on your calculator. Now, we just saw that whenever we have an exponential equation B to the power of X equals M we can rewrite this in its log form as log B of M is equal to X. Now, the same thing is true of anything with a base E, we treat it just like any other base. So if I have E to the power of X equals M, I can rewrite that in its log form as log base E of M is equal to X. Now, one thing we just need to be careful of here is that this log base E should always be rewritten as simply Ln. You're pretty much never going to see log base E written out like that. It's always going to be written as that Ln. So with that in mind, let's walk through these examples together. Now our first example here is X is equal to the natural log Ln of 17. So here let's go ahead and actually rewrite this as log base E to help this make a little bit more sense to us. So here we have X is equal to log base E because that's what the natural log is of 17. And we want to rewrite this in its exponential form because it's currently a log. So we're going to start the same place we always do with that base. So here our base is E. So starting with that base and then circling to the other side of my equal sign ee to the power of X and then circling back to my original side of my equal sign is equal to 17. And that's my final answer here. E to the power of X equals 17, just like we would any other base. So let's look at one final example here we have E to the power of X equals four and we want to rewrite this in its log form. So again, we're going to start at the same place with our base. So here this base of E becomes the base of my log. So log base E circle to the other side of your equal sign log base EF four and then circle back is equal to X. Now, we're not quite done here because remember we need to make sure we have the correct notation going on in our final answer. So we know that this log base E is really just L and the natural log. So this is really Ln A four equals X. And here's my final answer here. So now that we know a little bit more about the natural log, let's get a bit more practice. Thanks for watching and let me know if you have questions.

Change the following logarithmic expression to its equivalent exponential form.

$\log_4x=5$

• A. $4^{x}=5$
• B. $x^4=5$
• C. $4^5=x$
• D. $5^4=x$

Change the following logarithmic expression to its equivalent exponential form.

$x=\log9$

• A. $x=9$
• B. $9^{x}=10$
• C. $1^{x}=9$
• D. $10^{x}=9$

Change the following exponential expression to its equivalent logarithmic form.

$3^{x}=7$

• A. $\log_37=x$
• B. $\log_73=x$
• C. $\log_3x=7$
• D. $\log7=3^{x}$

Change the following exponential expression to its equivalent logarithmic form.

$e^9=x+3$

• A. $\log\left(x+3\right)=9$
• B. $\ln\left(x+3\right)=9$
• C. $\ln9=x+3$
• D. $\log_9x=e^3$

Evaluate Logarithms

Hey, everyone, we just learned that a log is the inverse of an exponential. But who cares? And how is that going to help us? Well, you're going to be asked to evaluate logs like this log base two of the cube root of two without using a calculator. And looking at that, you might think it's impossible to get an answer there without using a calculator. But here I'm going to show you that this log base two of the cube root of two is actually just equal to one third. And we can get that answer simply using the fact that the log is the inverse of an exponential. So with that in mind, let's go ahead and get started. Now, we're going to look at a couple of different properties here and the name of these properties is not important just that we know how to use them. So let's get started with our first one, the inverse property. Now, here we see log base two of two to the power of three. And the first thing you might notice here is that the base of my log and the base of my exponent are the same now, whatever that happens, these are simply going to cancel out, leaving me with just that three. It's kind of similar to if I take the square root of something squared, I'm just left with that something. This is the same idea here. Now, the same thing happens if I take an exponent of some base and raise it to a log of that same base, those are also going to cancel just leaving me with whatever is left over in this case three. And this is because logs and exponentials of the same base are always going to cancel because they are each other's inverse. Now this will happen no matter what the base is. If I take a log base E of E to the power of X, I would still simply be left with X or if I took E and raised it to the power of log base E as long as those bases are the same, it doesn't matter, they're going to cancel, leaving me with whatever is left over. So let's look at another property here here I have log base two of two. Now, here we see that the base of our log is the same thing that we're taking the log of. And we can think about this kind of similarly to our previous property where this two is actually the same thing as two to the power of one because two to the power of one is just two. Now with that in mind using our inverse property, I know that this log of base two and my exponent of base two are going to cancel leaving me with just one. Now, this is actually going to work for any log of any base if I take the log of some base and I'm taking the log of that same number, that is the base. I'm simply going to be left with one because taking the log of its base always equals one because our inverse property. Now let's look at one final property here, here I have log base two of one. Now, here it's going to be helpful to think about this in its exponential form. So if I take my log of base two and I think about what power I need to raise it to in order to equal one. So two to the power of X equals one. I just want to think about what number I could plug in for X that would actually give me this one. So I know that two to the power of one is two. So it's not that and I know that two squared is four. So I actually need a lower number. But if I take it to two to the power of zero, I know that two to the power of zero, like anything to the power of zero is one. So here, my answer would simply be zero and this will be true anytime we take the log of any base of one. So any log of one is always going to be equal to zero no matter what. Now with these properties in mind, let's take a look at some examples down here. Now looking at our first example, we already know what the answer is, but let's figure out how to get to that answer. So log base two of the Q root of two, I want to think about how I can rewrite this in a way that something is going to cancel. Now, this Q root of two, I know that I can actually rewrite this as an exponent because the QE root of two is actually just two to the power of one third using our exponent rules to make that QE route into an exponent. So this is really log two of two to the power of one third. Now, with that in mind, using our inverse property, I know that there's a log base two and to cancel simply leaving me with one third, which is my answer, no calculator needed. Now, let's move on to our next example here we have the natural log of one. Now I know that the natural log is really log base E so it's still a log and I know that any time I take the log of any base of one, I'm simply going to end up with zero. So that's my answer here. Just zero. Let's look at another example here, we have a log of 10. Now, log by itself is the common log. So this is really a log base 10 of 10. And since the base is the same as what I'm taking the log of using this second property up here, I know that this is simply going to be equal to one. And I'm done here. Now, we have one final example here, we have this log base five of 1/5 and looking at this first glance, I'm not really sure exactly. I'm going to get an answer here. So we're gonna have to be a little bit clever sometimes in thinking about how we can manipulate this in order to get something to cancel and get a final answer. Now, since I have this log base five, I know that I probably want my base of my exponent or what I'm taking the log of to be five. So how can I rewrite this 1/5 year? Well, I know that if I have a fraction, I could rewrite it as that number to the power of negative one. So this is really log base five of five to the power of negative one. So we had to be a little bit clever there. But now that we're here, we can go ahead and just cancel some stuff out. So this log base five and five cancel leaving me with just that negative one, which is my final answer. Negative one. So now that we know how to evaluate logs with no calculator needed just using some inverse properties. Let's get some more practice.

Evaluate the given logarithm.

$\log_77^{0.3}$

• A. 1.79
• B. 7
• C. 1
• D. 0.3

Evaluate the given logarithm.

$\frac32\log1$

• A. $\frac32$
• B. 0
• C. 1
• D. 10

Evaluate the given logarithm.

$\log_9\frac{1}{81}$

• A. 81
• B. 2
• C. – 2
• D. 9