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Okay, we're gonna have our first of our three videos on the topic of logarithms. This is a topic that often confuses people, so we're gonna go through this very carefully. Supposed we start with an exponential equation, n equals b to the k. And I'll just say right now, n is called the power, b is called the base, and k is called the exponent.

And so when it's written in this form, this equation is already solved for n, it's already solved for the power. And there comes up the question, how would we solve this equation for the other variables? Now, if we're adding, if we add a = b+c, of course we undo additions by subtraction.

And if we were multiplying, a = b times c, we undo multiplication by division. And notice that addition and multiplication are both commutative, that means we can switch the order around. a + b = b + a, a times b = b times a, and so it means we could subtraction or division to find either term. Things get a little complicated when we get to exponentiation because exponentiation Is not commutative, b to the k does not equal k to the b.

In general, those are two different things. And so, it means we need a different procedure to solve for b, or to solve for k. Well, solving for b, this is something we've already learned, we would just take a root. In fact, we'd take the kth root, if we wanted to solve for b.

b equals the kth root of n, so that's the form solved for the base. We use roots to solve for the base. I will say on the test, it's a unlikely scenario that you'd have a lot of questions about higher order roots. It is something the test could ask, it's an unlikely topic. That still leaves the question of how can we solve for the exponent.

And there's no simple algebraic way, there's no algebraic rearranging we can do, so we can get an equation k equals. So to address this, mathematicians use logarithms. And technically they have invented logarithms very specifically to address this particular problem, solving an equation for the exponent. That's why logarithms were invented.

And so the basic definition, is that, N equals b to the k, is equivalent to log base b of N equals k. And again, let me write that out, the equation on the right is read, log base b of N equals k. And so, these two are true, by definition. And in fact these two contain exactly the same information.

So if you're given one you can write it in terms of the other. The one on the left is called exponential form, the one on the right is called logarithmic form, and they contain exactly the same information. And so it's the same as if you had an equation say y equals 1 over x, or x equals 1 over y, Those two equations, they're just algebraic range but they contain the same information.

It's really just the same equation written in a different form. In much the same way these two are really the same equation written in a different form. And notice that when we write it in logarithmic form, that allows us to solve for the exponent, that equals the exponent. So let's think about some concrete examples, all right?

So all four of those, those are four exponential equations, they're 100% true. And what we'd like to do is write each one, just rewrite it, as a logarithmic equation, okay? So, 7 squared equals 49. Seven is the base, two is the exponent, 49 is the power. And so that's gonna be log base 7 of the power 49 equals the exponent 2.

The second one is going to be log of the base 10, log base 10, of the power, which is 10,000, equals the exponent 4. The third one is gonna be log base 3 of the power 27 = 3. The fourth one is going to be log base 5 of 125th, and that = -2. Okay, so what's going on here?

We can rewrite these in logarithmic form, but what is actually going on here? So a few things to notice about logarithms. So first of all, those equations, the exponential equation, n = b to the k, and the logarithmic equation, log base b of n equals k, those are two different forms of each other. So they contain precisely the same mathematical information.

And again, it's just as if we had x + y = 4, or y = 4- x. All we've done is we rearranged the same equation. So these are not two different things, it's the same mathematical fact rewritten in another form. And then very important, fundamentally, a logarithm is an exponent. If you understand only one thing about logarithms, understand that sentence, a logarithm is an exponent.

In particular, the logarithm, log base b of n is the exponent we would have to give to b to get an output of n. And so in other words if we start with b, and gave it an exponent of log base b of n, then the power that it would equal would be n. That's a really fundamental idea about logarithms. And we're gonna express that algebraically also.

So for example, suppose we start with that exponential form that solves for the power, that solve for N. So suppose what we do is we plug that in, w e replace the N in the logarithmic equation with b to the k, and what we get is log base b of b to the k = k. Well, think about what that says. What we're saying is, what exponent would we have to give to b, to get b to the k?

Well, of course to get b to the k, the exponent we'd have to give it is k, and so that's what that equation is really saying. And so, for example, if we had to find log base 3 of 3 to the 12, well, what exponent do we have to give to a base of 3, to get 3 to the 12. Of course we'd have to give 12. And so that equation is not only a handy shortcut to know, it's really a very deep idea to understand.

And there's a corresponding shortcut also, we can start with the logarithmic form, which is solved for k. And we can substitute that into the exponential form, and what we get is N = b to the power of log base b of N. And in some sense you could say this is the core definition of what a logarithm is, expressed in algebraic form.

And so for example, if someone asked us to find 5 to the power of log base 5 of 7. So this looks like a horrible thing, this looks like something where you need a gigantic calculation to figure out the value. But think about what it says, think about that exponent. Log base 5 of 7 bu definition is the exponent we would have to give to 5 to get an output of 7.

Well here we're taking that special exponent and we're giving it to 5. And so of course we're gonna get an output of 7. And so 5 to the power of log base 5 of 7 = just 7. And so it's really important, not just to memorize the two algebraic formulas on this page, but really to understand why they are summarizing in a profound way the fundamental definition of a logarithm.

So here's a practice problem. Pause the video, and then we'll talk about this. Okay, so let's look at that equation and the 1-x, that's pretty easy, but log base 5, of 5 to the x. So what are we really saying there? What exponent would we have to give to 5, to get a power of 5 to the x?

Well of course, we would have to give it an exponent of x. So log base 5 of 5 to the x = x. So that allows us to simplify enormously the equation. Well now we've replaced that expression with x, we get a ridiculously easy equation, okay? Add x to both sides, divide by 1, x = 1/2.

So if we understand the fundamental definition of a logarithm, then this becomes a really easy equation to solve. We know x = 1/2, we choose answer choice D. In summary, if b to the k equals N, then log base b of N equals k. Those are two different ways to write precisely the same mathematical information.

And so, it's very important to get comfortable switching back and forth between one and the other, because they contain the same information. A logarithm, fundamentally, is an exponent, that's really important. That is the core idea of a logarithm. And in particular, these two algebraic forms are different ways, and really, all we're doing is we're plugging one equation into the other.

We're doing that two different ways. But then, algebraically, we're coming out with a statement that summarizes that fundamental information. Log base b of k means, what exponent will we have to give to b To get a power, b to the k, of course, we'd have to give it k. And then, log base b of N, is the exponent we have to give to b, to get an output of N.

So when we give it to b, we get an output of N. And so, it's really important to understand what's going on in those equations. Not just memorize them, but understand them as algebraic representations of this fundamental definition.

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