Logarithms, Explained - Steve Kelly

对数全攻略 - Steve Kelly

翻译人员: Yoshinari Fukuzawa 校对人员: Cissy Yun

How does the difference between

你知道下面两个数字的差异

point 0000000398 and

0.0000000398

point 00000000398

和 0.00000000398

cause one to have red eyes after swimming?

会导致有人游泳后眼睛变红吗？

To answer this, we first need a way of

要回答之前，我们先要有个方法

dealing with rather small numbers,

or in some cases extremely large numbers.

来处理很小或很大的数字

This leads us to the concept of logarithms.

这带出了「对数」的观念 (log 也有「木材」的意思)

Well, what are logarithms?

Let's take the base number, b,

嗯，什么是对数？

我们把 b 当作底数 然后取它的 p 次方

and raise it to a power, p,

like 2 to the 3rd power,

就像 2 的 3 次方那样

and have it equal a number n.

结果等于数字 n

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

就得到一个指数方程式： b 的 p 次方等于 n

In our example, that'd be 2 raised

在这个例子里 就是2 的 3 次方等于 8

to the 3rd power, equals 8.

The exponent p is said to be

所以说指数 p就是n的对数

the logarithm of the number n.

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”

这堆变数让人开始头昏眼花了

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

so let's show this with an example.

我们用个例子来解释

What is the value of

log以10作为底数的10,000等于？

log base 10 of 10,000?

The same question could be asked using exponents:

这问题也可以用指数的方式来问

"10 raised to what power is 10,000?"

10 的几次方是 10,000？

Well, 10 to the 4th is 10,000.

10 的 4 次方是 10,000 因此以 10 为底 10,000 的对数

So, log base 10 of 10,000

must equal 4.

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 用工程计算机来算

Log base 10 is used so frequently

以 10 为底的对数在科学里太常用了

in the sciences

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

所以大多数计算机都有这个按钮

If the calculator will figure out logs for me,

如果计算机可以帮我算对数

why study them?

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

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？

and need to understand base 2?

So what is log base 2 of 64?

比如说以 2 为底 64 的对数是多少？

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

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

这跟游泳后，只在某些泳池

in some swimming pools

and not others?

我的眼睛会变红，有什么关系？

Well, it leads us into an interesting

这就关系到 对数在化学里的妙用了：

use of logarithms in chemistry:

finding the pH of water samples.

计算溶液的 pH 值

pH tells us how acidic or basic a sample is,

pH 值会告诉我们溶液的酸碱性

and can be calculated with the formula:

公式是 pH = - log 10 [H+]

pH equals negative log base 10 of the hydrogen ion concentration, or H plus.

其中 [H+] 是氢离子浓度

We can find the pH of water samples

我们可以轻易用计算机算出 氢离子浓度

with hydrogen ion concentration of point 0000000398

0.0000000398

and point 00000000398

和 0.00000000398

的 pH 值

quickly on a calculator.

Punch: negative log of each of those numbers,

按下去： 这两个数取对数再加负号

and you'll see the pH's are 7.4 and 8.4.

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,

因为泪水的 pH 值大约是 7.4

the H plus concentration of .0000000398

所以氢离子浓度是 7 个 0 及 398 的溶液，对我们眼睛刚刚好

will feel nice on your eyes,

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

但 pH 8.4 就会让你的眼睛又红又痒

对数很好记： 以 b 为底 n 的对数是 p

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?"

相当于在想： 「b 的几次方会是 n？」

"The BASE raised to what POWER equals the NUMBER?"

「b 的几次方会是 n？」 「b 的几次方会是 n？」

So now we know logarithms are very powerful

所以现在你了解到对数的威力

when dealing with extremely small or large numbers.

Logarithms can even be used

在于处理很大或很小的数字

instead of eyedrops after swimming.