You probably learned about them in your high school math class and then never thought about them again. I admit, to this day, logarithms (logs for short) are not my best friend. They’re not that intuitive to think about. But they are really useful, so I’ve learned to grudgingly embrace them.

您可能在中学数学课上就了解了它们，然后再也没有考虑它们。 我承认，到目前为止，对数(简称对数)不是我最好的朋友。 他们不是那么直观的思考。 但是它们确实很有用，所以我学会了勉强拥抱它们。

定义 (Definition)

A logarithm is the answer to the question what power x do I need to apply to the base b in order to obtain the number y:

对数是该问题的答案，我需要对底数b施加多少次幂才能获得数y ：

log_b(y) = xis another way of specifying the relationship:b^x = y

Let’s plug in some numbers to make this more clear. We will do base-10, so b=10.

让我们插入一些数字以使其更清楚。 我们将以10为底，因此b = 10。

log_10(100) = 2The base-10 logarithm of 100 is 2 because:10^2 = 100

It’s basically asking how many b’s do I need to multiply together to get y. To get 100, I just need to multiply two 10s together.

基本上是在问我需要乘多少个b才能得到y。 要获得100，我只需要将两个10乘在一起即可。

This is actually a really nice way of dealing with multiplicative sequences.

实际上，这是处理乘法序列的一种非常好的方法。

Assume something is growing at a changing rate denoted by rn. Over 3 years, its total growth rate is:(1+r1)*(1+r2)*(1+r3)In log-scale, we would just take the log of the entire thing:(I will use log to denote log_10 to simplify notation)One rule of logs is that log(A*B*C) = log(A) + log(B) + log(C) so:log{(1+r1)*(1+r2)*(1+r3)}= log(1+r1) + log(1+r2) + log(1+r3)Now let's think about log(1+r1) - it's asking what power do I need to apply to 10 so that it equals (1+r1):10^z1 = 1+r1, so log(1+r1) = z110^z2 = 1+r2, so log(1+r2) = z210^z3 = 1+r3, so log(1+r3) = z3This allows us to rewrite each term, and the previous equation simplifies to:log(1+r1) + log(1+r2) + log(1+r3)= z1 + z2 + z3Thus, in log-space, a multiplicative sequence becomes an additive one:log{(1+r1)*(1+r2)*(1+r3)} = z1 + z2 + z3

一个例子 (An Example)

The average non-math immersed person is most likely to run into logs on graphs that plot in log-scale. For example, here’s a log-scale (base-10) graph of the total U.S. COVID-19 case count:

一般的非数学沉浸式人员最有可能在以对数比例绘制的图形上遇到对数。 例如，这是美国COVID-19病例总数的对数刻度(以10为底)图：

It always takes me a second to orient myself when I see a log-scale graph. What do the numbers on the y-axis mean? What does it mean when the plot starts to flatten out and plateau?

当我看到对数刻度图时，总是花一秒钟的时间来调整自己的方向。 y轴上的数字是什么意思？ 情节开始趋于平坦并达到平稳状态是什么意思？

Let’s take a look at the raw un-logged data for total U.S. case counts. So we can more easily compare, I plotted the un-logged data in the top chart and the logged version (same as the previous plot) in the bottom chart.

让我们看一下未记录的原始数据，了解美国的总病例数。 因此，我们可以更轻松地进行比较，在顶部的图表中绘制了未记录的数据，在底部的图表中绘制了已记录的版本(与之前的图相同)。

There’s not a whole lot we can glean from the top subplot (the unlogged data) besides that cases have increased rapidly going from 0 to almost 6 million in just over 200 days. But what we really want to know is whether the rate of infection is declining, and it’s impossible to tell that using the unlogged data. In fact, because of the scaling (and the apparent flatness of the unlogged data in the beginning), an inexperienced analyst viewing this chart might wrongly conclude that the rate of infection was very low during the first 50 days before it suddenly accelerated.

除了上面的案例在短短200天内从0Swift增加到近600万，我们无法从顶层子图中收集到很多信息。 但是，我们真正想知道的是感染率是否正在下降，并且无法使用未记录的数据来判断这一点。 实际上，由于规模的扩大(以及开始时未记录数据的明显平坦性)，缺乏经验的分析师查看此图表可能会错误地得出结论，感染率在突然加速之前的前50天内非常低。

That would be completely opposite to reality. Log-scale to the rescue. I will explain how it works in a second, but just trust me for now when I say that:

那将与现实完全相反。 对数缩放以进行救援。 我将在一秒钟内解释它是如何工作的，但是当我这样说时，请暂时信任我：

• An increase in the slope of the log-scale line means that the rate of infection is increasing.对数刻度线的斜率增加意味着感染率正在增加。
• A constant slope means that the rate of infection is not changing.斜率恒定意味着感染率没有变化。
• And a decrease in slope means that the rate of infection is declining.斜率降低意味着感染率正在下降。
• The y-axis is powers of the base. So in this case since it’s base-10, 1 on the y-axis means 10¹, 2 on the y-axis means 10², etc.y轴是基数的幂。 因此，在这种情况下，由于它的基数是10，所以y轴上的1表示10¹，y轴上的2表示10²，依此类推。

Going back to our plot, we can see that the log-scale chart tells a completely different story than the unlogged one. The rate of infection was actually very high during the first 75 or so days before decelerating. So logarithms can quickly tell us whether the rate of change of something is increasing (like a car speeding up), staying constant, or decreasing (gradually stepping on the brake). Let’s use one more example to see how logarithms help us better eyeball the growth trend in our data. Let’s say we have data that grows at the following high but constantly declining rates:

回到我们的情节，我们可以看到对数刻度图讲述的故事与未记录的故事完全不同。 在减速之前的前75天左右，感染率实际上很高。 因此，对数可以快速告诉我们某些事物的变化率是在增加(例如汽车加速)，保持恒定还是在减少(逐渐踩刹车)。 让我们再举一个例子，看看对数如何帮助我们更好地了解数据的增长趋势。 假设我们的数据增长速度很高，但一直在下降：

rates = [1.5, 1.4, 1.3, 1.2, 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3]

From the un-logged plot (left), I honestly have no idea whether the growth rate is declining or not. But from the log-scale plot (right), it’s immediately obvious that growth is slowing.

从未记录的图(左)来看，老实说，我不知道增长率是否在下降。 但是，从对数刻度图(右)可以明显看出，增长正在放缓。

我们为什么在乎？ (Why Do We Care?)

For me, back when I was learning math, the part that was often missing was knowing why I should care about a specific concept. So in a practical sense, why do logs matter?

对我来说，当我学习数学时，经常缺少的部分是知道为什么我应该关心一个特定的概念。 因此，从实际意义上讲，日志为何如此重要？

The answer (and there are others but this is the most practical one) is that logs allow us much more flexibility when dealing with things that grow exponentially. And contrary to popular belief, exponential growth doesn’t always have to be like a tree growing into the sky — it merely means that the data is compounding (or decaying) at a reasonably constant rate. Some examples of exponential growth are:

答案(还有其他答案，但这是最实用的答案)是，在处理指数级增长的事物时，日志使我们具有更大的灵活性。 与流行的看法相反，指数增长并不一定总是像生长在空中的树一样–只是意味着数据以合理的恒定速率复合(或衰减)。 指数增长的一些示例是：

• A savings account. It might not pay much these days, but it’s still compounding constantly. A savings account that pays 2% a year would double your money in 35 years (1.02³⁵ = 2). It’s exponential because to get the total growth, we take 1 plus the growth rate and raise it by the power of N, where N is the number of years.

  储蓄账户。 这些天可能付出不了多少，但仍在不断增加。 每年支付2％的储蓄帐户将在35年内使您的资金翻一番(1.02³= 2)。 这是指数级的，因为要获得总增长率，我们将增长率乘以1，然后乘以N的幂将其提高，其中N是年数。

• Inflation. Conversely every year, we lose 2% in purchasing power (goods and services become 2% more expensive). Our money, if not invested, loses 50% of its value in 35 years (0.98³⁵ = 0.5).通货膨胀。 相反，每年，我们的购买力下降2％(商品和服务的价格上涨2％)。 如果不进行投资，我们的钱将在35年内损失其价值的50％(0.98³= 0.5)。
• An obvious one is a pandemic like the one that we’re in. In its early days back in February and March of this year, infections were growing at an average rate of 20% a day. So for a time, we were doubling COVID-19 cases basically every 4 days (1.2⁴ = 2.07)! That’s part of the reason why it was and is so imperative to carefully track and bring down the rate of infection.一种明显的流行病就是我们所处的流行病。在今年2月和3月的早期，感染的平均增长率为每天20％。 因此，有一段时间，我们基本上每4天就将COVID-19病例增加一倍(1.2⁴= 2.07)！ 这就是为什么现在必须如此，要认真跟踪并降低感染率的原因。
• The cost of college has grown about 8% a year. That’s good for a double every 9 years (1.08⁹ = 2).大学的费用每年增长约8％。 这是每9年翻一番的好结果(1.08 good = 2)。
• Moore’s law states that the amount of transistors on a microchip doubles roughly every 2 years, which implies an annual growth rate of 41%.摩尔定律指出，微芯片上的晶体管数量大约每两年增加一倍，这意味着年增长率为41％。

As a general rule of thump, if we can say something approximately doubles (or halves) every N years, then we have exponential or compounding growth.

作为重击的一般规则，如果我们每隔N年就可以说大约翻一番(或减半)，那么我们的增长就是指数级的或复合的。

For fun, let’s plot all of these (besides COVID-19 because it would explode off the graph) first normally and then in log-scale. Here’s the code for the un-logged plot:

为了好玩，让我们首先正常地然后以对数比例绘制所有这些图(除了COVID-19以外，因为它会从图形上爆炸)。 这是未记录图的代码：

import numpy as npimport pandas as pdimport matplotlib.pyplot as pltyrs = 30rates = [0.02, -0.02, 0.08, 0.41]labels = ['Savings Account', 'Real Value of Cash',           'Cost of College', 'Transistors on Chip']values = []for r in rates:    temp_list = []    for y in range(yrs+1):        temp_list.append((1+r)**y)    values.append(np.array(temp_list).reshape(-1,1))ax = pd.DataFrame(np.concatenate(values, axis=1),                   columns=labels).plot(figsize=(10,6));ax.set_ylabel('Growth factor with starting value of 1')ax.set_xlabel('Years')plt.tight_layout()

The plot is not very interesting. Transistors grow at such a high rate that they make everything else look like a flat line.

剧情不是很有趣。 晶体管以如此高的速度增长，使其他所有东西看起来像一条平线。

Let’s see if we can get a better view of what’s going on with a log-scale graph. To get log-scale on the y-axis, we just toggle the logy option in Pandas’ plot method (which plots the y-axis in log-scale):

让我们看看是否可以通过对数刻度图更好地了解正在发生的事情。 为了在y轴上获得对数刻度，我们只需在Pandas的plot方法中切换logy选项(该方法在对数刻度上绘制y轴)：

ax = pd.DataFrame(np.concatenate(values, axis=1), columns=labels).plot(figsize=(10,6), logy=True);ax.set_ylabel('Growth factor with starting value of 1 (log base 10 scale)')ax.set_xlabel('Years')plt.tight_layout()

The log-scale graph is four straight lines. Why’s that? The short answer is they are all straight lines because they are all increasing (or decreasing in the case of the Real Value of Cash) at a constant rate.

对数刻度图是四个直线。 为什么？ 简短的答案是它们都是直线，因为它们都以恒定的速率增加(或在现金实际价值的情况下减少)。

Let’s take a deeper look using Moore’s law, which states that transistor count should grow at 41% a year:

让我们使用摩尔定律进行更深入的研究，该定律指出晶体管的数量应该以每年41％的速度增长：

          Unlogged          log_10Year 0:   1 transistor      0      <- 10^0    = 1Year 1:   1.41              0.15   <- 10^0.15 = 1.41Year 2:   1.98              0.30   <- 10^0.30 = 1.98Year 3:   2.80              0.45   <- 10^0.45 = 2.80Year 4:   3.95              0.60   <- 10^0.60 = 3.95Explanation:We get year 3’s value of 2.80 as 1.41^3.Because 10^0.15 = 1.41, we can reframe this as:1.41^3 = (10^0.15)*((10^0.15)*(10^0.15)Now let's take the log of both sides using the rule that:log(A*B*C) = log(A) + log(B) + log(C)We get:(I will use log to denote log_10 to simplify notation)log(1.41^3) = log{(10^0.15)*((10^0.15)*(10^0.15)}log(1.41^3) = log(10^0.15) + log(10^0.15) + log(10^0.15)Recall log(10^0.15) asks what power do I need to apply to 10 in order to get the value in the parentheses, 10^0.15? Obviously the answer is 0.15.So the previous equation simplifies to:log(1.41^3) = 0.15 + 0.15 + 0.15 log(1.41^3) = 0.45Thus, something growing 41% a year, would in log-scale (with a base of 10), would grow by adding 0.15 a year like so:0, 0.15, 0.30, 0.45, 0.60And to get the actual unlogged values, you just apply these values as powers to the base:10^0, 10^0.15, 10^0.30, 10^0.45, 10^0.60

对数线性化指数数据 (Logs Linearize Exponential Data)

Logs allow us to translate a multiplicative (a.k.a. compounding) sequence into an additive one. Besides making graphs of exponential data easier to interpret, this has another really useful application — linear regression. One of the requirements of linear regression is that the relationship between the dependent variable and the independent variable must be a linear one. Let’s make some fake data to visualize this:

对数允许我们将乘法(也称为复合)序列转换为加法序列。 除了使指数数据的图形更易于解释之外，它还有另一个非常有用的应用程序-线性回归。 线性回归的要求之一是因变量和自变量之间的关系必须是线性的。 让我们制作一些伪造的数据来可视化这一点：

rates = [0.3, 0.8, 0.2, 0.1, 0.4, 0.7, 0.5, 1.2, 0.3, 0.1, 0.4, 0.9]x = [i for i in range(len(rates))]y = (np.array(rates)+1).cumprod()fig, (ax1, ax2) = plt.subplots(1,2,figsize=(14,6), sharex='all')ax1.scatter(x, y);ax1.set_xlabel('X');ax1.set_ylabel('Y');ax1.set_title('Raw Values');ax2.scatter(x, np.log10(y));ax2.set_xlabel('X');ax2.set_ylabel('Y');ax2.set_title('Log-Scale Values');plt.tight_layout()

Our y variable is growing at very high rates. And though the rate varies, it appears that as x gets bigger, y increases at a faster rate. Because the slope of the left plot is changing (and increasing), y and x do not share a linear relationship. So linear regression is a no go.

我们的y变量以非常高的速度增长。 尽管速率有所不同，但随着x变大，y会以更快的速率增加。 由于左图的斜率在变化(并且在增加)，因此y和x不共享线性关系。 因此线性回归是行不通的。

But once we apply a logarithm (base-10), we get the plot on the right. That looks pretty linear to me. Exponential data is exponential because with each time-step (assuming time series), the current value gets multiplied by some value:

但是，一旦我们应用对数(以10为底)，就可以在右侧得到图。 对我来说，这看起来很线性。 指数数据是指数的，因为对于每个时间步长(假设时间序列)，当前值都会乘以某个值：

At time step 1, y is: x^1At time step 2, y is: x^2At time step 3, y is: x^3

So there’s an exponential relationship, between y and x. But notice that there’s a linear relationship between y and the exponent that is applied to x. Taking the log (let’s cheat and use a base of x) gets us the exponent:

因此，y和x之间存在指数关系。 但是请注意， y和应用于x的指数之间存在线性关系 。 取日志(让我们作弊并使用x的底数)可以得到指数：

At time step 1: log(x^1) = 1At time step 2: log(x^2) = 2At time step 3: log(x^3) = 3

So now we have the linear relationship between log(y) and x that we need for linear regression.

因此，现在我们有了线性回归所需的log(y)和x之间的线性关系。

结论 (Conclusion)

And that’s all folks. Logs are an integral part of math and this post only scratches the surface of how useful they can be. But hopefully after reading this, you no longer find them scary and are encourages to use them in your daily work! Cheers!

那就是所有人。 日志是数学不可或缺的一部分，这篇文章只是从表面上说明了日志的用途。 但是希望阅读此书后，您不再觉得它们令人恐惧，并鼓励他们在日常工作中使用它们！ 干杯!