好的，我们将在Logarithms主题上拥有我们的三个视频。这是一个经常混淆人的主题，所以 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.

因此，当它以这种形式写入时，该等式已经解决了n，它已经为电源解决了。 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, 这意味着我们可以在周围切换订单。a + b = b + a，次b = b次a，等等 这意味着我们可以减法或分割以找到任一期限。当我们开始指数时，事情会变得有点复杂，因为 指数不是换向的，B到k不等于k到b。

一般来说，这些是两种不同的东西。因此，这意味着我们需要一个不同的程序来解决 b，或解决k。好吧，解决B，这是我们已经学到的东西， 我们只会拿一个根。事实上，如果我们想解决b，我们就会拿走犹太根。

b equals the kth root of n, so that's the form solved for the base. We use roots to solve for the base. 我会在考试中说，这是一个不太可能的情景，你有很多关于更高阶根的问题。 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. 所以要解决这个问题，数学家使用对数。在技​​术上，他们非常具体发明了Logarithms 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, 或x等于y，这两个方程式，它们只是代数范围，但它们包含相同的信息。

它真的只是用不同形式写的相同的等式。与之相同的方式，这两个是写入的相同等式 一种不同的形式。并注意我们以对数形式写入时， that allows us to solve for the exponent, that equals the exponent. So let's think about some concrete examples, all right?

所以所有四个，这些都是四个指数方程式，它们是100％的真实。我们想做的是写下每个人， just rewrite it, as a logarithmic equation, okay? So, 7 squared equals 49. 七是基础，两个是指数，49是力量。因此，POWER 49的日志基础7是指数2。

第二个将是基本10的日志，LOG基础10的功率，电力为10,000，等于指数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.

再一次,这只是如果我们有x + y = 4,或y= 4- x. All we've done is we rearranged the same equation. 所以这些不是两个不同的东西，它是以另一种形式重写的相同数学事实。 And then very important, fundamentally, a logarithm is an exponent. If you understand only one thing about logarithms, 了解该句子，对数是指数。

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. 这是关于对数的一个非常基础的想法。而且我们也会表达这代数。

So for example, suppose we start with that exponential form that solves for the power, that solve for N. 因此，假设我们所做的是我们将其插入，W e将具有b的对数方程中的n替换为k，以及我们得到的是b的log基础b到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. 因此，例如，如果我们不得不找到3到12的日志基础3，那么，我们必须给予3个基础的指数，以获得3到12。 Of course we'd have to give 12. And so that equation is not only a handy shortcut to know, 理解这真是一个非常深刻的想法。

并且还有一个相应的快捷方式，我们可以从对数形式开始，该标语被解决。 我们可以将其替换为指数形式，我们得到的是n = b到n的日志基础b的力量。 在某种意义上，您可以说这是对数的核心定义，以代数形式表示。

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 省略了巨大的计算。但想想它所说的话，想到这一点。 7 BU定义的日志基数5是我们必须给予5的指数，以获得7的输出。

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. 所以5到日志基础的力量为7 =只是7.所以它非常重要，而不仅仅是记住两个代数 公式在此页面上，但真的要理解为什么他们总结了对数的基本定义。

所以这是一个练习问题。暂停视频，然后我们会谈谈这个。 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. 那么我们真的在那里说什么？我们必须给予5个指数，让5到x的力量？

当然，我们必须给它一个x指数。所以将Log Base 5为5到x = x。 因此，允许我们简化了巨大的等式。那么现在我们已经用x替换了表达式， 我们得到了一个易易易上的等式，好吗？将X添加到两侧，除以1，x = 1/2。

So if we understand the fundamental definition of a logarithm, then this becomes a really easy equation to solve. 我们知道x = 1/2，我们选择答案选择d。总之，如果b到k等于n，则n等于k的日志b基础b。 这些是两种不同的方式来写入相同的数学信息。

因此，在一个与另一个之间来回切换是非常重要的，因为它们包含相同的信息。 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.

我们正在做两种不同的方式。但后来，代数， 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, 得到n的输出。

所以当我们把它交给b时，我们得到了n的输出，了解那些正在进行的事情非常重要 equations. Not just memorize them, but understand them as algebraic representations of this fundamental definition.