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究竟什么是对数

Logarithms, Explained - Steve Kelly

它们之间的不同有哪些?
How does the difference between
0.0000000398和
point 0000000398 and
3.98e-09
point 00000000398
使人在游泳后眼睛发红
cause one to have red eyes after swimming?
为了回答这个问题,我们要先找到一种方法来
To answer this, we first need a way of
处理非常小的数字
dealing with rather small numbers,
或者在某些情况下非常大的数字。
or in some cases extremely large numbers.
这个让我们有对数这个概念。
This leads us to the concept of logarithms.
嗯啊,那什么是对数呢?
Well, what are logarithms?
我们先设置底数b,
Let’s take the base number, b,
然后给他一个指数,p
and raise it to a power, p,
就像2³一样。
like 2 to the 3rd power,
让它等于数字n。
and have it equal a number n.
我们得到一个指数等式:b的p次方等于n。
We get an exponential equation: b raised to the p power, equals n.
在我们举的例子中,2
In our example, that’d be 2 raised
自乘了3次,等于8。
to the 3rd power, equals 8.
指数p可以说成
The exponent p is said to be
数字n的对数。
the logarithm of the number n.
大多数时候将会这样写:
Most of the time this would be written:
log,底数b,数字,等于p,指数。
"log, base b, of a number equals p, the power."
现在这些变量听起来让人有些困惑,
This is starting to sound a bit confusing with all the variables,
因此让我们再举个栗子。
so let’s show this with an example.
以10为底,10000的
What is the value of
对数是多少?
log base 10 of 10,000?
这个问题同样可以用指数表示
The same question could be asked using exponents:
10的多少次方是10000?
"10 raised to what power is 10,000?"
嗯啊,10的4次方是10000。
Well, 10 to the 4th is 10,000.
所以,以10为底10000的对数
So, log base 10 of 10,000
等于4。
must equal 4.
科学计算器可以很简单的计算出这个结果。
This example can also be completed very simply on a scientific calculator.
以10为底的对数经常
Log base 10 is used so frequently
用在科学上
in the sciences
在大多数计算器上面也有这个单独的按钮。
that it has the honor of having its own button on most calculators.
如果计算器可以帮我们算出对数,
If the calculator will figure out logs for me,
那为什么我们还要学习它?
why study them?
就提示一点:
Just a quick reminder:
log键只能计算以10为底的对数。
the log button only computes logarithms of base 10.
如果你想进一步知道科学计算器
What if you want to go into computer science
和以2为底的对数?
and need to understand base 2?
以2为底64的对数是多少?
So what is log base 2 of 64?
换句话说,2的多少次方是64?
In other words, 2 raised to what power is 64?
嗯啊,我们来数一下。2,4,8,16,32,64
Well, use your fingers. 2, 4, 8, 16, 32, 64.
所以以2为底64的对数肯定是6。
So log base 2 of 64 must equal 6.
嗯啊,那么做什么让你眼睛变红
So what does this have to do with my eyes turning red
在一些游泳池里
in some swimming pools
而不是其他的?
and not others?
嗯啊,当对数用来化学上会发生
Well, it leads us into an interesting
有趣的事:
use of logarithms in chemistry:
举个有关水的pH值的例子。
finding the pH of water samples.
pH值是告诉我们水的酸碱性的样本,
pH tells us how acidic or basic a sample is,
它可以通过公式计算:
and can be calculated with the formula:
pH值等于以10为底氢离子或H+浓度的对数的负数。
pH equals negative log base 10 of the hydrogen ion concentration, or H plus.
我们可以用计算器快速算出水样本中
We can find the pH of water samples
氢离子浓度为0.0000000398
with hydrogen ion concentration of point 0000000398
和0.00000000398
and point 00000000398
的pH值是多少。
quickly on a calculator.
按键:负的log加上这些数字
Punch: negative log of each of those numbers,
你将看到pH值是7.4和8.4。
and you’ll see the pH’s are 7.4 and 8.4.
我们眼中的泪水PH值大约是7.4,
Since the tears in our eyes have a pH of about 7.4,
H+的浓度是0.70398时
the H plus concentration of .70398
对你的眼睛来说刚刚好,
will feel nice on your eyes,
但是PH值达到8.4时,会让你的眼睛又痒又红。
but the pH of 8.4 will make you feel itchy and red.
对数很容易记住,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"
底数的多少次方等于n?
by repeating: "The base raised to what power equals the number?"
底数的多少次方等于n?底数的多少次方等于n?
"The BASE raised to what POWER equals the NUMBER?"
现在我们知道了,对数对于
So now we know logarithms are very powerful
处理特别小或者相当大的数字时是非常方便的。
when dealing with extremely small or large numbers.
对数甚至可以用来代替游泳后用的眼药水。
Logarithms can even be used instead of eyedrops after swimming.

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