Let's quickly review exponential equations. These equations are exponential since the unknown varibale was the exponent. What would x be in this case.
Here x would equal positive 4. We would just rewrite 81 as 3 to the fourth, so we can set the exponents equal to one another since the bases are the same. Great work if you found 4.
So we know, this is the fourth power. 3 to the 4th equals 81. And this form is considered to be exponential form. Now, another new form that we're going to talk about is logarithmic form. Log form involves a logarithm, and it's just another way of writing this equation. The key idea for a logarithm is that its base will be the same base in the equation above. So if we have 3 to the fourth equals 81, then we'll have log with a base of 3 set equal to 4. The answer to a logarithm is an exponent. Meaning if I take the log base 3 of some number, I'll get 4. Now it turns out that that number has to be 81. So we read this statement as log base 3 of 81 equals 4. We're asking ourselves, what's the exponent that we need to raise a base of 3 to, to get 81? Well, that exponent is 4. When we take the logarithm of any number, we'll always get an exponent. We use logarithms as a way to undo or reverse raising a number to a power. For example, if we had this exponential form, we can also write it in logarithmic form. This base becomes the base of our logarithm. So, we'll have log base two of sum number equals 3, the exponent. We know that this number must be 8, since we need to raise 2 to in order to get 8. So, in this case, log base 2 of 8 equals 3. In general, when we have something written in exponential form with a base raised to an exponent equal to a number, we can rewrite this in logarithmic form. We can make the base be the base of our logarithm, the number is the argument for a logarithm and the answer is the exponent. So log with base b of some number always will give us an exponent. I think the key idea in any of these situations is that the log of a number results in an exponent. The same was true for this example.
We've already seen a lot of logarithms used in this lesson and we're only going to keep using them more. So I think it's time that you learn how to use mathquill to write logs in a pretty way. Now they should be pretty similar to some other functions you've seen written in mathquill. And of course you'll get plenty of practice using this. Let's say for example that I want to write log base 4 of x. Well, in mathquill here is what I would type in, \ log, then the spacebar, then to get down into this subscript space where the four is written, need to press Shift and then hold it down while you type the Minus sign key. That's what's going to shift you down here. Then you can write a little four just to the bottom right of the log. To get out of the space and get back to where we're going to write the input for the log press the Arrow key to the right. You could also press Tab instead then take your input which in this case is x. Now since you've seen one example I think it would make sense for you to try one yourself. How would you type log base 3 of x plus 4 in mathquill. In the box that appears down here. Try it out. You should be able to tell based on how it looks, whether or not it's going to be right. You could also use the box down here to try out writing some other log expressions, anything you can think of.
Hopefully you were able to type this expression into MathQuill properly. But just in case it's giving you trouble, here are the keys you need to press to make it happen. To write log base 3 of x plus 4, we use something pretty similar to what we did up here. You type \ log and then space to type the word log. To move down to type the 3 as a subscript, hold down Shift while you press the button with the minus sign. Then you can enter your 3 and move back up here, press the Right Arrow key. They may need a parenthesis, an opening one. And then we have our input, x plus 4. Of course, MathQuill closes this parenthesis for you on its own. So you don't have to technically type another one, although you can if you want to. As I said before, you'll get a ton of practice typing in things like this in MathQuill in future lessons.
So what I'd like you to do is write this equation in logarithmic form
This can be written as log base 5 of 25 equals 2. Nice thinking if you found this. Remember that the base for this equation becomes the base of our logarithm, and that the logarithm of any number Is an exponent. This means log base five of 25 must equal 2. We're asking ourselves, what do we raise five to in order to get 25? That power is 2.
Let's see if you can do another one. This time with a variable. What do you think this equation would be written in logarithmic form?
Now, the key is we weren't really solving for x here. We know x equals 2 the base of 2, will be the base of our logarithm. So log base 2 of x equals 4, our exponent. If you always remember that this base becomes the base of the logarithm. And that this power becomes the answer to our logarithm, then you you know that this must be the argument of the logarithm. This is the number that we take the logarithm of. So 2 to the fourth equals x is the same thing as log base 2 of x equal to 4.
So in general, we know that the logarithm of some number is really an exponent. We can take the log of any number a, so long as this argument a is greater than greater than zero and not equal to 1. If b were 1, then a would have to be 1. And x could be anything since 1 raised to any power always equals 1. And just like before we see that our base is the base of our log rhythm. And the exponent is the answer to our logarithm. And a is the number that we take the logarithm of.
For this question let's see if you can go in the reverse direction. This is in logarithmic form and I want you to write it in exponential form. What do you think that looks like.
In exponential form, this would be 2 the fourth power equals 16. Great thinking if you got this equation. Keep in mind that our base on our logarithm is the base for our exponent. And the answer to a logarithm are the result of taking a logarithm of a number as an exponent, so this 4 is the exponent for 2. And then this should make sense, the 16 must go here since 2 to the fourth equals 16. Now another way to think about it,is going in the reverse direction. So, I'm going to right log base 2 of 16 equals 4 down here. We're in logarithmic form, so we want to send this base back up to be the base, and we want to send exponent back to its position. So, we'll have base of 2, raised to the fourth power. So, this is how we get 2 to the fourth equal to 16. We've just reversed the directions of the arrows from the first diagram that we saw. Either way you do it, you'll still get the correct answer. Just keep in mind your concept of which one's the base and then which one's the exponent. The result of a logarithm is always an exponent.
Now, let's get in some practice. I want you to write each of these equations in logarithmic form. You should have one logarithm, an equals sign and then the right numbers in the right spots. Good luck. Also, be sure that you don't use commas when entering any numbers. Especially for this number, 100,000. You don't want to put a comma, right here.
Here are the three solutions written in logarithmic form. Nice work, if you got all three right. Now, some of these were tough, so it's okay if you didn't get all of them right. Let's walk through each of them. For the first one, we'll have our base be 10. And we'll have our answer be 5. When we take the log with base 10 of some number, we'll get 5. It means that this number must be a in order to get a 100000. For the second one we'll use the same process. We have a base of 4 so this is the logarithm with a base 4. The result of our logarithm equals negative 3. Its the exponent and this is one sixty fourth here at the argument for our logarithm. It's the number that we take the logarithm of. And finally, for the last one, we use the same technique. Our base of our logarithm is 2, since this base is 2. The result of our logarithm is x. This is the exponent. And we're taking log base 2 of the number 3 to get x. Now, keep in mind, in this last equation, we weren't really solving for x. We were just trying to write this in logarithmic form. It turns out that this number x is really equal to log base 2 of 3. Now the logarithm in your calculator can't get this number. The log in your calculator has a base of 10. But there are other ways to get bases of 2. We won't cover that in this course, but if you're interested, do some research.
And finally for the last practice I want you write each of these in exponential form. You should have a base raised to a power set equal to some number. Good luck for each of these.
Here are the three equations. Great work if you found each of these. We'll start by making this base be the base for our exponent. So 3 is written down here. The answer of a logarithm is an exponent. So, we know this 2 is going to be the exponent for our base 3. And this should make perfect sense since 3 squared equals 9. This is that number. For the second one, we'll repeat the same process. The base of our logarithm is 5, so 5 will be the base for our exponent. The answer of a logarithm is an exponent. So this 3 is the exponent for our base of 5. And of course 5 to the third equals 125. And finally we'll have a base of one fifth here, an exponent of negative 2 which equals 25. Remember that this negative exponent will flip over or do the reciprocal of this base. So, one fifth will become 5 and then we'll square it to get 25. Nice work if you got each of these right. Even if you got two of them right, fantastic.