Logarithms, Explained - Steve Kelly

對數完全攻略 - 史蒂夫凱利 (Steve Kelly)

譯者: Jephian Lin 審譯者: Allen Li

How does the difference between

你知道下面兩個數字的差異

point 0000000398 and

0.0000000398

point 00000000398

和 0.00000000398

會導致有人游泳後眼睛變紅嗎？

cause one to have red eyes after swimming?

To answer this, we first need a way of

要回答之前，我們先要有個方法

dealing with rather small numbers,

來處理很小或很大的數字

or in some cases extremely large numbers.

This leads us to the concept of logarithms.

這帶出了「對數」的觀念 (log 也有「木材」的意思)

Well, what are logarithms?

Let's take the base number, b,

嗯，什麼是對數？

我們把 b 當作底數 然後取它的 p 次方

and raise it to a power, p,

like 2 to the 3rd power,

就像 2 的 3 次方那樣

and have it equal a number n.

結果等於數字 n

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

就得到一個指數方程式： b 的 p 次方等於 n

In our example, that'd be 2 raised

在這個例子裡，就是 2 的 3 次方等於 8

to the 3rd power, equals 8.

我們就說 n 的對數是 p

The exponent p is said to be

the logarithm of the number n.

通常我們會寫作 log 以 b 為底數的 n 等於 p

Most of the time this would be written:

"log, base b, of a number equals p, the power."

這堆變數讓人開始頭昏眼花了

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

so let's show this with an example.

我們用個例子來解釋

What is the value of

以 10 為底 10,000 的對數是多少？

log base 10 of 10,000?

The same question could be asked using exponents:

這問題也可以用指數的方式來問

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

10 的幾次方是 10,000？

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

嗯，10 的 4 次方是 10,000 因此以 10 為底 10,000 的對數

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.

這個例子也可以很簡單的 用工程計算機來算

Log base 10 is used so frequently

以 10 為底的對數 在科學裡太常用了

in the sciences

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

所以很光榮 大多數計算機都有這個按鈕

If the calculator will figure out logs for me,

如果計算機可以幫我算對數

why study them?

那我幹嘛學？

Just a quick reminder:

the log button only computes logarithms of base 10.

這要提醒一下 對數按鈕只能算以 10 為底的對數

What if you want to go into computer science

如果你想進入電腦科學的世界 用到以 2 為底的對數怎麼辦？

and need to understand base 2?

So what is log base 2 of 64?

比如說以 2 為底 64 的對數是多少？

換句話說， 2 的幾次方是 64？

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

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

嗯，用手指數一下 2, 4, 8, 16, 32, 64

以 2 為底，64 的對數就是 6

So log base 2 of 64 must equal 6.

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

這跟游泳後，只在某些泳池

我的眼睛會變紅，有什麼關係？

in some swimming pools

and not others?

Well, it leads us into an interesting

這就關係到 對數在化學裡的妙用了：

use of logarithms in chemistry:

finding the pH of water samples.

計算溶液的 pH 值

pH tells us how acidic or basic a sample is,

pH 值會告訴我們溶液的酸鹼性

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+] 是氫離子濃度

We can find the pH of water samples

我們可以輕易用計算機算出 氫離子濃度

with hydrogen ion concentration of point 0000000398

0.0000000398

和 0.00000000398

and point 00000000398

的 pH 值。按下去：

quickly on a calculator.

Punch: negative log of each of those numbers,

這兩個數取對數再加負號 你會得到 pH 值是 7.4 和 8.4

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

因為淚水的 pH 值大約是 7.4

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

所以氫離子濃度是 7 個 0 及 398 的溶液，對我們眼睛剛剛好

the H plus concentration of .0000000398

will feel nice on your eyes,

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

但 pH 8.4 就會讓你又紅又癢

對數很好記： 以 b 為底 n 的對數是 p

It's easy to remember logarithms "log base b of some number n equals p"

相當於在想： 「b 的幾次方會是 n？」

by repeating: "The base raised to what power equals the number?"

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

「b 的幾次方會是 n？」 「b 的幾次方會是 n？」

So now we know logarithms are very powerful

所以現在你了解到對數的威力

when dealing with extremely small or large numbers.

在於處理很大或很小的數字

對數甚至可在游泳後取代眼藥水！

Logarithms can even be used

instead of eyedrops after swimming.