0-0-0-0-0-0-0-3-9-8

0-0-0-0-0-0-0-3-9-8

and point 0-0-0-0-0-0-0-0-3-9-8

和0-0-0-0-0-0-0-0-3-9-8點。

cause one to have red eyes after swimming?

使人游泳後眼睛發紅？

To answer this, we first need a way of dealing with rather small numbers,

要回答這個問題，我們首先需要一個處理相當小的數字的方法。

or in some cases extremely large numbers.

或在某些情況下數量極多。

This leads us to the concept of logarithms.

這就引出了對數的概念。

Well, what are logarithms?

那麼，什麼是對數？

Let's take the base number - b - and raise it to a power, p,

讓我們'把基數--b--提高到一個冪，p。

like 2 to the 3rd power

像2到3倍的力量

and have it equal a number n.

並讓它等於一個數字n。

We get an exponential equation b raised to the p power equals n.

我們得到一個指數方程b提高到p冪等於n。

In our example, that'd be 2 raised to the 3rd power equals 8.

在我們的例子中，那'd是2提高到3次方等於8。

The exponent p is said to be the logarithm of the number n.

指數p被稱為是數字n的對數。

Most of the time this would be written "log base b of a number equals p, the power."

大多數情況下，這會被寫成"一個數的對數基數b等於p，冪.&quot。

This is starting to sound a bit confusing with all the variables,

這開始聽起來有點混亂，所有的變量。

so let's show this with an example.

所以讓我們'用一個例子來說明。

What is the value of log base 10 of 10 thousand?

萬的對數基數10的值是多少？

The same question could be asked using exponents.

同樣的問題可以用指數來問。

10 raised to what power is 10 thousand?

10提高到什麼力量是1萬？

Well, 10 to the 4th is 10 thousand. So, log base 10 of 10 thousand

好吧，10到4是1萬。所以，對數基數10的萬

must equal 4.

必須等於4。

This example can also be completed very simply on a scientific calculator.

這個例子也可以在科學計算器上非常簡單地完成。

Log base 10 is used so frequently in the sciences

對數基數10在科學中使用頻率很高。

that it has the honor of having its own button on most calculators.

它有幸在大多數計算器上擁有自己的按鈕。

If the calculator will figure out logs for me,

如果計算器能幫我算出日誌。

why study them?

為什麼要研究它們？

Just a quick reminder, the log button only computes logarithms of base 10.

提醒一下，日誌按鈕只計算10基的對數。

What if you want to go into computer science and need to understand base 2?

如果你想進入計算機科學領域，需要了解基數2怎麼辦？

So what is log base 2 of 64?

那麼64的對數基數2是什麼呢？

In other words, 2 raised to what power is 64?

換句話說，2提高到什麼功率是64？

Well, use your fingers. 2, 4, 8, 16, 32, 64.

好吧，用你的手指。2, 4, 8, 16, 32, 64.

So log base 2 of 64 must equal 6.

所以64的對數基數2一定等於6。

So what does this have to do with my eyes turning red

這跟我眼睛變紅有什麼關係？

in some swimming pools and not others?

在一些游泳池中，而不是在其他游泳池中？

Well, it leads us into an interesting use of logarithms in chemistry:

好吧，這讓我們進入了一個有趣的化學中對數的應用。

finding the pH of water samples.

尋找水樣的pH值。

pH tells us how acidic or basic a sample is,

pH值告訴我們樣品的酸鹼度。

and can be calculated with the formula pH equals negative log base 10

並可以用公式計算出pH值等於負對數鹼10

of the hydrogen ion concentration, or H plus.

的氫離子濃度，即H加。

We can find the pH of water samples with hydrogen ion concentration of

我們可以求出氫離子濃度為的水樣的pH值為

point 0-0-0-0-0-0-0-3-9-8

0-0-0-0-0-0-3-9-8點

and point 0-0-0-0-0-0-0-0-3-9-8

和0-0-0-0-0-0-0-0-3-9-8點。

quickly on a calculator. Punch:

迅速在計算器上。衝。

negative log of each of those numbers, and you'll see the pHs are 7.4 and 8.4.

這些數字的負對數，你'會看到pH值是7.4和8.4。

Since the tears in our eyes have a pH of about 7.4,

由於我們眼睛裡的淚水的pH值約為7.4。

the H plus concentration of .70398 will feel nice on your eyes.

H加濃度為0.70398，眼睛會感覺很舒服。

But the pH of 8.4 will make you feel itchy and red.

但pH值為8.4，會讓你覺得很癢，很紅。

It's easy to remember logarithms - log base b of some number n

這很容易記住對數--某個數字n的對數基數b。

equals p - by repeating "the base raised to what power equals the number?"

等於p--通過重複"基數提高到多少倍等於多少?&quot。

The base raised to what power equals the number? The base raised to what power equals the number?

基數提高到多少倍等於多少？基數提高到多少倍等於多少？

So now we know logarithms are very powerful

所以現在我們知道了對數是非常強大的。

when dealing with extremely small or large numbers.

當處理極小或極多的數字時。

Logarithms can even be used instead of eyedrops after swimming.

游泳後甚至可以用對數代替眼藥水。