Introduction to Logarithms: Study with Video Lessons, Practice Problems & Examples

To solve exponential equations, we can use logarithms as the inverse operation. For example, if \( b^x = m \), it can be rewritten as \( \log_b(m) = x \). The natural logarithm, denoted as \( \ln \), is specifically for base \( e \). Key properties include \( \log_b(b) = 1 \) and \( \log_b(1) = 0 \). Understanding these concepts allows for evaluating logs without a calculator, enhancing problem-solving skills in mathematics.

Logarithms Introduction

Hey, everyone. Whenever we solve polynomial equations like x3=216, we can simply do the reverse operation in order to isolate x. So here, since x is being 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 if my variable x is instead in my exponent, like this 2x=8? How are we going to isolate x here? Well, I know that looking at this, I can just think, okay, how many times does 2 need to be multiplied by itself in order to give me 8? And I know that my answer is simply x is equal to 3. But what if I'm given something like 2x=216? I really don't want to have to multiply 2 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 2 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 log of 2x, and that's equal to log of 216. But we actually do need to consider one more thing here because whenever we canceled the 3 on our x3, we took the cube root. We didn't take the square root or the fourth root or anything else. We took the cube root in order to cancel that 3. 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 2, I want my log to have a base of 2 as well. So really, I want to take a log base 2 of 2x and then on the other side, a log base 2 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 log base 2 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 2 of 216. Well, 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? Well, looking at our function here or our equation here, log base 2 of 216, this is really saying, Okay, what power does 2 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, 2x=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, 3x=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 start at the same place. We're going to start with our base. So this base 3 of our exponent is going to become the base of our log. So I start here with log base 3. 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 3 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 3 of 81 is going to equal, so that's equal to x. Now I have my equivalent statement in its log form. Log base 3 of 81 equals x is equivalent to 3x=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 4 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 4. That becomes the base of my exponential. So I have a log base 4, and I have that 4. And now I want to raise 4 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 4x and then finally circling back to the other side of our equal sign, that is our final number here, 64. So 4x=64 is an equivalent statement in exponential form from this x equals log base 4 of 64. Now I know that that might seem like 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 5 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 5, so I'm going to start with an exponential of base 5. Now once I have that base, I'm going to circle to the other side of my equal sign and get that x. So 5x, 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 5x=800 is my equivalent statement in its exponential form. Let's look at another example here. We have log base 2 of 16 equals 4. Now where are we going to start here? Well, we want to start with our base, of course, so I have this log base 2 so that becomes the base of my exponent. So I have 2 and then I go to the other side of my equal sign 2 to the power of 4 and then a circle back to where you started. 2x=16. Now this is great because we can actually see that this is a true statement. 24=16, so I know that I wrote that correctly. Let's look at one final example here. Here we have 10x=4500. So this is in its exponential form. We wanna go ahead and put this in log form. 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 base 10 of 4500 is equal to x and that is my equivalent statement. But looking at this log base 10 of 4500, this log base 10 is actually a sort of special type of log. So log base 10 is actually known as the common log because it occurs so frequently. So it can actually just be written as 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 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 special notation. It's always written as ln. 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 bx=m, we can rewrite this in its log form as logb(m)=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 ex=m, I can rewrite that in its log form as loge(m)=x. 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. Our first example here is x=ln(17). Here, let's go ahead and actually rewrite this as log base e to help make this a little bit more sense to us. So here we have x=loge(17). And we want to rewrite this in its exponential form because it's currently a log. So we're going to start at the same place we always do with that base. So here our base is e. Starting with that base and then circling to the other side of my equal sign, ex=17, and then circling back to my original side of my equal sign is equal to 17. And that's my final answer here, ex=17, just like we would any other base.

Let's look at one final example here. We have ex=4, and we want to rewrite this in its log form. 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 loge(4)=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 ln, the natural log. So this is really ln(4)=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 log223 without using a calculator. And looking at that, you might think it's impossible to get an answer there without using a calculator, but I'm going to show you that this log223 is actually just equal to 13. 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. Let's get started with our first one, the inverse property. Now here we see log223. 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, whenever that happens, these are simply going to cancel out, leaving me with just that 3. 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, 3. 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 logeex, I would still simply be left with x. Or if I took eloge, 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 log22. Now here, we see that the base of our log is the same thing that we're taking the log of. Using our inverse property, knowing that this log of base 2 and my exponent of base 2 are going to cancel, I am left with just 1. 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 1 because taking the log of its base always equals 1 due to our inverse property.

Now let's look at one final property here. Here I have log21. Now here, it's going to be helpful to think about this in its exponential form. So if I take my log of base 2 and I think about what power I need to raise it to in order to equal 1, so 2x=1, I just want to think about what number I could plug in for x that would actually give me this one. I know that 20, like anything to the power of 0, is 1. So here, my answer would simply be 0. And this will be true any time we take the log of any base of 1. So any log of 1 is always going to be equal to 0 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 log223. I want to think about how I can rewrite this in a way that something is going to cancel. Now, this cube root of 2, I know that I can actually rewrite this as an exponent because the cube root of 2 is actually just 213 using our exponent rules to make that cube root into an exponent. So this is really log2213. Using our inverse property, I know that this log base 2 and 2 cancel, simply leaving me with 13, which is my answer, no calculator needed.

Now let's move on to our next example. Here we have the natural log of 1. 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 1, I'm simply going to end up with 0. So that's my answer here, just 0.

Now let's look at another example here. We have log of 10. Now log by itself is the common log, so this is really log1010. 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 1 and I'm done here.

Now we have one final example here. We have this log515. Now, looking at this at first glance, I'm not really sure exactly how I'm going to get an answer here. So we're going to 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. 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 log55-1. 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 5 and 5 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

Here’s what students ask on this topic:

What is a logarithm and how is it related to exponents?

A logarithm is the inverse operation of an exponent. If you have an exponential equation like $b^x=m$, it can be rewritten in logarithmic form as ${\log }_b\left(m\right)=x$. This means that the logarithm gives you the power to which the base must be raised to get the number. For example, ${\log }_2\left(8\right)=3$ because $2^3=8$.

How do you convert between exponential and logarithmic forms?

To convert from exponential to logarithmic form, start with the base of the exponent. For example, if you have $b^x=m$, it converts to ${\log }_b\left(m\right)=x$. Conversely, to convert from logarithmic to exponential form, start with the base of the log. For example, ${\log }_b\left(m\right)=x$ converts to $b^x=m$.

What are the properties of logarithms?

Key properties of logarithms include:
1. ${\log }_b\left(b\right)=1$ because any number raised to the power of 1 is itself.
2. ${\log }_b\left(1\right)=0$ because any number raised to the power of 0 is 1.
3. ${\log }_b\left(b^x\right)=x$ because the log and the exponent cancel each other out.

What is the natural logarithm and how is it different from common logarithms?

The natural logarithm, denoted as $\ln$, is a logarithm with base $e$ (approximately 2.718). It is written as $\ln$ instead of ${\log }_e$. The common logarithm, denoted as $\log$, has a base of 10. Both follow the same properties but are used in different contexts. For example, $\ln \left(1\right)=0$ and $\log \left(1\right)=0$.

How do you evaluate logarithms without a calculator?

To evaluate logarithms without a calculator, use properties of logarithms. For example, ${\log }_2(2^{\frac{1}{3}}$ because the log and exponent cancel out. Similarly, $\ln \left(1\right)=0$ because any log of 1 is 0. For ${\log }_5\left(\frac{1}{5}\right)=-1$, rewrite $\frac{1}{5}$ as $5^{-}$ and use the property to get $-1$.