How does the difference between point 0000000398 and point 00000000398 cause one to have red eyes after swimming?
你知道下面两个数字的差异，0.0000000398和0.00000000398，会导致有人游泳后眼睛变红吗？
To answer this, we first need a way of dealing with rather small numbers, or in some cases extremely large numbers.
在回答之前，我们先要有个方法来处理很小或很大的数字，
This leads us to the concept of logarithms. Well, what are logarithms?
这就引出了对数的概念。那么什么是对数呢？
Let's take the base number, b, and raise it to a power, p, like 2 to the 3rd power, and have it equal a number n.
我们把b当作底数，然后取它的p次方，就像2的3次方那样，结果等于数字n。
We get an exponential equation: b raised to the p power, equals n.
然后我们就得到了一个指数方程式：b的p次方等于n。
In our example, that'd be 2 raised to the 3rd power, equals 8.
在这个例子里，就是2的3次方等于8。
The exponent p is said to be the logarithm of the number n.
所以说指数p就是n的对数。
Most of the time this would be written: 'log, base b, of a number equals p, the power.'
通常我们会写作，“log以b为底数的n等于p”。
This is starting to sound a bit confusing with all the variables, so let's show this with an example.
这堆变量让人开始头昏眼花了，那么我们就用个例子来解释。
What is the value of log base 10 of 10,000?
log以10作为底数的10000等于多少？
The same question could be asked using exponents: '10 raised to what power is 10,000?'
这问题也可以用指数的方式来问，10的几次方是10000？
Well, 10 to the 4th is 10,000. So, log base 10 of 10,000 must equal 4.
10的4次方是10000。因此以10为底10000的对数就是4。
This example can also be completed very simply on a scientific calculator.
这个例子也可以很简单的用科学计算器来算。
Log base 10 is used so frequently in the sciences that it has the honor of having its own button on most calculators.
以10为底的对数在科学里太常用了，所以大多数计算器都有这个按钮。
If the calculator will figure out logs for me, why study them?
如果计算器可以帮我算对数，那我干嘛学？
Just a quick reminder: the log button only computes logarithms of base 10.
这要提醒一下：对数按钮只能算以10为底的对数。

What if you want to go into computer science and need to understand base 2?
如果你想进入电脑科学的世界，用到以2为底的对数怎么办？
So what is log base 2 of 64? In other words, 2 raised to what power is 64?
比如说以2为底64的对数是多少？换句话说，2的几次方是64？
Well, use your fingers. 2, 4, 8, 16, 32, 64.
嗯，用手指数一下。2, 4, 8, 16, 32, 64。
So log base 2 of 64 must equal 6.
以2为底，64的对数就是6。
So what does this have to do with my eyes turning red in some swimming pools and not others?
这跟游泳后只在某些泳池我的眼睛会变红有什么关系？
Well, it leads us into an interesting use of logarithms in chemistry: finding the pH of water samples.
这就关系到对数在化学里的妙用了：计算溶液的pH值。
pH tells us how acidic or basic a sample is, and can be calculated with the formula:
pH值会告诉我们溶液的酸碱性公式是：
pH equals negative log base 10 of the hydrogen ion concentration, or H plus.
pH=-log10(H+)，其中(H+)是氢离子浓度。
We can find the pH of water samples with hydrogen ion concentration of point 0000000398 and point 00000000398 quickly on a calculator.
我们可以轻易用计算器算出氢离子浓度0.0000000398和0.00000000398的pH值。
Punch: negative log of each of those numbers, and you'll see the pH's are 7.4 and 8.4.
按下去：这两个数取对数再加负号，你会得到pH值是7.4和8.4。
Since the tears in our eyes have a pH of about 7.4, the H plus concentration of .0000000398 will feel nice on your eyes,
因为我们眼中的泪水的pH值大约是7.4，所以氢离子浓度是0.0000000398的溶液对我们眼睛刚刚好，
but the pH of 8.4 will make you feel itchy and red.
但pH8.4就会让你的眼睛又红又痒。
It's easy to remember logarithms 'log base b of some number n equals p' by repeating: 'The base raised to what power equals the number?'
通过重复'b的几次方会是n'，以此来记住以b为底n的对数是p是很容易的。
'The BASE raised to what POWER equals the NUMBER?'
b的几次方会是n？
So now we know logarithms are very powerful when dealing with extremely small or large numbers.
所以现在你了解到，对数在处理很大或很小的数字时，是非常强大的。
Logarithms can even be used instead of eyedrops after swimming.
对数甚至可以在游泳后取代眼药水！