Exponential & Logarithmic Functions
AP Precalculus Topic 2 14:05 English narration · English + 中文 subtitles burned in
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Transcript
Here are two ways for a quantity to grow.
有两种让一个量增长的方式。
The blue line grows steadily, adding the same amount every step.
蓝色的直线稳定增长，每一步加上相同的量。
The other one is doubling — multiplying by the same factor every step.
另一条则在翻倍—— 每一步乘以相同的倍数。
At first the doubling one is smaller, and it looks like it will lose.
起初翻倍的那条更小，看起来会输。
But watch.
但看好了。
It races past the line, and after that the gap only widens.
它冲过了那条直线， 之后差距只会越来越大。
Repeated multiplication always wins in the end — and that is the power of exponential growth.
反复相乘最终总会胜出——这就是指数增长的威力。
Welcome to Unit Two: exponential and logarithmic functions.
欢迎来到第二单元：指数与对数函数。
We begin with sequences and growth by multiplication, meet the exponential function and its rules, then turn it around to discover its inverse — the logarithm.
我们从数列和"以相乘方式增长"讲起，认识指数函数 及其运算法则，再把它反过来，发现它的反函数——对数。
Let's begin.
让我们开始吧。
Start with sequences, because they are the discrete version of everything that follows.
先从数列讲起，因为它们是后面一切内容的离散版本。
A sequence is a function from the whole numbers to the reals, so its graph is DISCRETE POINTS, not a curve — say that if you are asked.
数列是从自然数到实数的函数，所以它的图像是一些离散的点，而不是曲线—— 如果被问到，就要这样答。
An arithmetic sequence has a common difference d, a constant rate of change: a n equals a nought plus d n, or from any known term, a k plus d times n minus k.
等差数列有公差 d，也就是恒定的变化率： a n 等于 a 零加 d n，或者从任何已知项出发，等于 a k 加 d 乘以 n 减 k。
A geometric sequence has a common ratio r, a constant PROPORTIONAL change: g n equals g nought times r to the n.
等比数列有公比 r，也就是恒定的比例变化：g n 等于 g 零乘以 r 的 n 次方。
Here is the sentence that captures the difference.
下面这句话抓住了两者的区别。
An increasing arithmetic sequence grows by the same AMOUNT each step; an increasing geometric one grows by a LARGER amount each step.
递增的等差数列每一步增加的量相同； 而递增的等比数列每一步增加的量越来越大。
Put numbers on it.
代上数字看看。
An arithmetic sequence with a nought equal to three and d equal to five gives a four equals three plus five times four, which is twenty-three.
等差数列取 a 零等于三、d 等于五， 则 a 四等于三加五乘以四，等于二十三。
A geometric sequence with g nought equal to two and r equal to three gives g four equals two times three to the fourth, which is one hundred and sixty-two.
等比数列取 g 零等于二、r 等于三， 则 g 四等于二乘以三的四次方，等于一百六十二。
They started within one of each other and after only four steps the geometric one has pulled far ahead — seven times larger.
它们起点只差一，而仅仅四步之后，等比的那个就遥遥领先了——大了七倍。
That gap is the whole reason exponential models matter, and it only widens.
这个差距正是指数模型之所以重要的全部原因，而且它只会越拉越大。
Two kinds of change sit at the heart of this unit.
两种变化方式是本单元的核心。
An arithmetic sequence grows by repeated addition — add the same common difference every step, like three, eight, thirteen, eighteen.
等差数列以反复相加的方式增长——每一步加上相同的公差， 比如三、八、十三、十八。
Its continuous cousin is the linear function.
它的连续版本是线性函数。
A geometric sequence grows by repeated multiplication — multiply by the same common ratio every step, like two, six, eighteen, fifty-four.
等比数列以反复相乘的方式增长—— 每一步乘以相同的公比，比如二、六、十八、五十四。
Its cousin is the exponential function.
它的版本是指数函数。
So these are the two cousins we build on: add for linear, multiply for exponential.
所以我们依靠这两个亲戚： 相加对应线性，相乘对应指数。
An exponential function has the form: an initial value times a base raised to x.
指数函数的形式是：一个初始值乘以一个底数的 x 次方。
When the base is bigger than one, you get growth — the curve climbs faster and faster.
当底数大于一，你得到增长—— 曲线越升越快。
When the base is between zero and one, you get decay — the curve falls toward zero.
当底数在零和一之间，你得到衰减——曲线趋向于零。
Either way, the curve hugs a horizontal asymptote at y equals zero, and it is always heading the same direction, always bending the same way.
无论哪种，曲线都紧贴 一条水平渐近线 y 等于零，而且它始终朝同一方向、始终朝同一方向弯曲。
So an exponential has no extrema and no points of inflection.
所以指数函数 没有极值，也没有拐点。
Here is exponential decay in a picture: a fixed percentage is lost each period.
这是指数衰减的画面：每个时段损失固定的百分比。
The initial value a must not be zero, and the base b is positive and not equal to one.
初始值不能为零，底数必须为正且 不等于一。
Domain is all real numbers.
定义域是全体实数。
Over equal-length input intervals the outputs stay proportional — that is the signature of an exponential.
在等长的输入区间上，输出保持成比例——这就是指数的 特征。
So the curve is always increasing or always decreasing, and always concave the same way.
所以曲线总是单调增或单调减，弯曲方向始终不变。
Bacteria under a microscope are a classic model.
显微镜下的细菌是经典模型。
A population that multiplies by a constant factor each time step is exponential growth.
每一步乘以固定倍数的种群，就是指数增长。
The same idea fits car value that loses a fixed percent each year, or a dose of medicine that falls by a fixed fraction each hour.
同样的想法也 适合每年贬值固定百分比的车价，或每小时下降固定比例的药物剂量。
Whenever equal time steps multiply the quantity by the same ratio, reach for an exponential.
只要相等的时间 步都让量乘以同一个公比，就用指数模型。
Exponentials are ruled by the laws of exponents.
指数由指数运算法则支配。
The product rule: multiply two powers of the same base, and you add the exponents.
乘法法则：同底数的两个幂相乘，就把指数相加。
The power rule: raise a power to a power, and you multiply the exponents.
幂的乘方法则： 幂再乘方，就把指数相乘。
A negative exponent means one over the power — a reciprocal.
负指数表示幂的倒数。
And a unit-fraction exponent is a root — the k-th root.
而分母为一的分数指数就是方根——k 次方根。
These four rules reshape every exponential expression you will meet.
这四条法则能改写你会遇到的每一个指数表达式。
The same rules rewrite graph moves.
同样的法则也能改写图像变换。
A horizontal shift of the exponent — base to the power of x plus k — is the same as a vertical stretch of the whole exponential, with stretch factor equal to base to the k.
指数上的水平平移——底数的自变量加常数次方——等于整个 指数函数的竖直伸缩，伸缩因子是底数的该常数次方。
A horizontal stretch, base to the power of c times x, is just a change of base: the new base is the old base raised to c.
水平伸缩，底数的常数乘自变量次方， 只是换底：新底是旧底的该常数次方。
So algebra and transformations speak the same language.
所以代数与变换说的是同一种语言。
Exponentials model quantities that grow by a constant PROPORTION over equal intervals — repeated multiplication.
指数函数用来刻画在相等的间隔上按恒定比例增长的量——也就是反复相乘。
There are three ways to build one.
建立这样一个模型有三种途径。
From a ratio and an initial value, which gives b and a directly.
由一个比值和一个初始值出发，这直接给出 b 和 a。
Or from TWO points, where you solve the system for a and b.
或者由两个点出发，这时你要解方程组求 a 和 b。
Sometimes a constant must be ADDED to the data before the proportional pattern shows up, because the data sit on a shifted asymptote rather than on zero.
有时必须先给数据加上一个常数，比例关系才会显现出来， 因为数据所依托的是一条平移过的渐近线，而不是零。
And for a whole data set, use exponential regression on technology.
而对整个数据集，就在计算器上用指数回归。
One convention: the natural base e, about two point seven one eight, is the standard base for real-world models.
有一个惯例：自然底数 e，约等于二点七一八，是现实模型的标准底数。
When a rate of change shifts only slightly, linear, quadratic and exponential models may ALL seem to fit, and the exam asks you to choose.
当变化率只是略有改变时，线性、二次和指数模型可能看上去都拟合得不错， 而考试要求你作出选择。
Use two things: context, and fit quality.
用两样东西来判断：情境，和拟合的优劣。
For fit quality, a model is appropriate if its RESIDUAL PLOT shows no pattern — and a residual is actual minus predicted.
就拟合而言，如果一个模型的残差图看不出任何规律，它就是合适的—— 而残差等于实际值减去预测值。
Read that carefully.
这句话要仔细读。
It is not that the residuals should be small; it is that they should be PATTERNLESS.
要求的不是残差小，而是残差没有规律。
A curved band of residuals means the wrong shape of model, however small they are.
残差排成一条弯曲的带状，就说明模型的形状选错了，无论残差有多小。
Then the error is the gap between predicted and actual, and the CONTEXT decides whether an over-estimate or an under-estimate is the safer mistake.
然后，误差是预测值与实际值之间的差距， 而由情境来决定高估还是低估才是更安全的错误。
Before we invert a function, we need composition.
在把函数反过来之前，我们需要复合。
A composite function feeds one function's output into another: f of g of x.
复合函数把一个函数的输出送进另一个函数：f 括号 g 括号 x。
Think of two machines in a row — x goes into g, and g's output goes into f.
想象两台机器排成一排——x 进入 g，g 的输出再进入 f。
Always evaluate the inside first.
永远先算里面的。
And order matters: composition is not commutative, so f of g is usually different from g of f.
而且顺序重要： 复合不满足交换律，所以先 g 后 f 通常不同于先 f 后 g。
Think of composition like nested dolls — evaluate the inside first, then the outer layer.
把复合想成套娃——先算里面，再算外层。
To build f of g of x analytically, substitute g of x for every x in f.
要从解析式构造复合函数，就把外层里的每个自 变量换成内层的表达式。
The identity function, which just returns x, leaves any function unchanged under composition.
恒等函数只返回输入本身，和任何函数复合都不变。
A function can also be decomposed into simpler pieces: an additive shift is composing with x plus k, and a dilation is composing with k times x.
你也可以把 函数拆成更简单的片段：加法平移是与自变量加常数复合，伸缩是与常数乘自变量复合。
The domain of a composite is the inputs of g whose outputs land inside the domain of f.
复合函数的定义域是内层的那些输入，它们的输出落在外层的定义域里。
Picture a function as a machine: input goes in, output comes out.
把函数想成一台机器：输入进去，输出出来。
Its inverse runs the machine backwards — output becomes the new input and recovers the original.
它的反函数把机器倒着运行——输出变成新的 输入，还原原来的值。
A function is invertible on a domain where each output comes from a unique input.
函数在某个定义域上可逆，当且仅当每个输出只来自唯一输入。
If that fails, restrict the domain until every output has only one pre-image.
如果 做不到，就缩小定义域，直到每个输出只有一个原像。
Domain and range then swap under inversion, and composing either way gives the identity.
于是定义域与值域对调，两种复合 顺序都得到恒等。
An inverse function runs the machine backwards.
反函数把机器倒着运行。
If a function sends a to b, its inverse sends b back to a.
如果一个函数把 a 送到 b，它的反函数就把 b 送回 a。
On a graph, that means reflecting across the line y equals x — every point swaps its two coordinates.
在图上，这意味着关于直线 y 等于 x 作反射——每个点交换它的两个坐标。
To find an inverse's formula, swap x and y, then solve.
要求反函数的公式， 就交换 x 和 y，再解出来。
This reflection is the key to the logarithm.
这种反射正是通向对数的钥匙。
Here is the logarithm.
现在来看对数。
A logarithm answers one question: what exponent?
对数只回答一个问题：是几次方？
Log base b of c equals a means exactly that b raised to the a gives c.
以 b 为底 c 的对数等于 a，恰好表示 b 的 a 次方等于 c。
The logarithm simply pulls the exponent out into the open.
对数不过是把指数拉到明面上来。
For example, log base two of eight is three, because two to the third power is eight.
举个例子，以二为底八的对数是三， 因为二的三次方是八。
Reading a logarithm as an exponent is the whole trick.
把对数读成指数，就是全部诀窍。
So the logarithm is the inverse of the exponential.
所以对数是指数的反函数。
The curves y equals two to the x and y equals log base two of x are mirror images across the line y equals x.
曲线 y 等于二的 x 次方，与 y 等于以二为底 x 的对数， 关于直线 y 等于 x 互为镜像。
Because they undo each other, a log of a power gives back the exponent, and a base raised to a log gives back the number.
因为它们互相抵消，幂的对数会还原出指数，底数的对数次方 会还原出那个数。
The logarithmic function is defined only for positive inputs, and it has a vertical asymptote at x equals zero — the mirror of the exponential's horizontal one.
对数函数只对正数有定义，并且在 x 等于零处有一条竖直渐近线—— 正是指数函数那条水平渐近线的镜像。
Logarithms also model data that span huge multiplicative ranges — sound intensity, acid strength, earthquakes.
对数也用来给跨越巨大倍数范围的数据建模——声强、酸碱度、地震。
Build a log model from data, and use logarithmic regression when technology is allowed.
可以从数据建立对数 模型，在允许用技术时做对数回归。
Because a log compresses large values, it turns proportional growth into a straight-line pattern.
因为对数压缩大数值，它把比例增长变成直线模式。
That is the idea behind semi-log plots.
这正是半对数图背后的想法。
On a logarithmic scale, each unit is a multiplicative step of the base — ten to the zero, ten to the one, ten to the two, and so on.
在对数尺度上，每一格都是底数的一次相乘跨步——十的零次、 十的一次、十的二次，以此类推。
Because logs are exponents, the log laws mirror the exponent rules.
因为对数就是指数，对数法则与指数法则相对应。
The product law: the log of a product is the sum of the logs.
乘积法则：乘积的对数等于对数之和。
The quotient law turns division into subtraction.
商法则把除法变成减法。
The power law: the log of a power brings the exponent out in front as a multiplier.
幂法则：幂的对数把指数拿到前面作为系数。
And the change of base formula lets your calculator find any log, by dividing one natural log by another.
而换底公式让你的 计算器能算出任何对数——用一个自然对数除以另一个自然对数。
Now we can solve exponential equations, using the fact that logs and exponentials undo each other.
现在我们可以解指数方程了，利用对数与指数互相抵消这一点。
Take two times three to the x equals fifty-four.
看二乘以三的 x 次方等于五十四。
First, divide by two: three to the x equals twenty-seven.
先除以二：三的 x 次方等于二十七。
Twenty-seven is three cubed — the same base on both sides — so x equals three.
二十七是三的三次方——两边同底——所以 x 等于三。
When the sides are not tidy powers, take logs of both sides instead.
当两边不是整齐的幂时，就对两边取对数。
For five to the x equals twenty, x is the natural log of twenty over the natural log of five, about one point eight six.
对于五的 x 次方等于二十，x 等于二十的自然对数 除以五的自然对数，约等于一点八六。
Describe the graph of log base b of x precisely, because these are the words the rubric wants.
要准确描述以 b 为底 x 的对数的图像，因为评分标准要的就是这些词。
Its domain is x greater than zero and its range is all the reals — exactly the reverse of the exponential's.
它的定义域是 x 大于零，值域是全体实数——正好和指数函数反过来。
It is always increasing when b is bigger than one, always decreasing when b is between zero and one, and always concave the same way.
当 b 大于一时它恒为递增，当 b 在零和一之间时它恒为递减，而且凹向始终不变。
It has a VERTICAL asymptote at x equals zero, which is the mirror image of the exponential's HORIZONTAL asymptote at y equals zero.
它在 x 等于零处有一条竖直渐近线， 这正是指数函数在 y 等于零处那条水平渐近线的镜像。
And it grows very slowly for large x — which is precisely the property that makes it useful for compressing huge ranges.
而且它在 x 很大时增长得非常慢—— 而这恰恰就是它能用来压缩巨大范围的那个性质。
Your calculator only has two log buttons — base ten and base e — but questions use every base.
你的计算器上只有两个对数键——以十为底和以 e 为底—— 但题目用到的底数五花八门。
The change-of-base formula fixes that: log base b of x equals log x over log b, or equally l n x over l n b.
换底公式解决了这个问题： 以 b 为底 x 的对数，等于 log x 除以 log b， 也同样等于 l n x 除以 l n b。
Either base works, because the two conversions cancel in the ratio.
两种底都可以，因为在这个比值里两次换算相互抵消了。
So log base three of twenty is l n twenty over l n three.
所以以三为底二十的对数就是 l n 二十除以 l n 三。
One caution when writing it down: the base goes on the BOTTOM.
写的时候有一点要当心：底数放在分母上。
Getting that upside down gives you the reciprocal of the answer, and it is a common slip.
写反了得到的是答案的倒数，而这是常见的失误。
Two procedures, and one check that is worth a mark on its own.
有两套步骤，还有一项本身就值一分的检验。
To solve an exponential equation: isolate the exponential, then take a log of both sides, using the power property to bring the exponent down.
解指数方程：先把指数项单独放到一边，再对两边取对数， 并用幂的性质把指数拿下来。
To solve a logarithmic equation, do the reverse: isolate the log, then exponentiate both sides.
解对数方程则反过来：先把对数项单独放到一边，再对两边取指数。
Now the check.
现在说检验。
The argument of a log must stay POSITIVE, so once you have your answers you must substitute them back and DISCARD any that make a log's argument zero or negative.
对数的真数必须保持为正， 所以求出答案之后，你必须把它们代回去， 并舍去任何使某个对数的真数为零或为负的解。
Those are extraneous solutions.
这些就是增根。
They appear legitimately from the algebra, so nothing warns you — only the check catches them.
它们是代数运算合法地产生出来的，所以没有任何提示， 只有这一步检验能把它们抓出来。
One last tool: the semi-log plot.
最后一个工具：半对数图。
On normal axes, an exponential curves upward, bending more and more.
在普通坐标轴上，指数曲线向上弯，越弯越厉害。
Now change the vertical axis to a logarithmic axis.
现在把纵轴换成 对数轴。
On this log axis, the very same exponential becomes a perfectly straight line.
在这条对数轴上，同一个指数就变成一条完美的直线。
That is because taking the log of the equation gives log y equals log a plus log b times x — a straight line, whose slope is log b.
这是因为对方程取对数，得到 log y 等于 log a 加上 log b 乘以 x——一条直线，斜率是 log b。
So if data look straight on a semi-log plot, an exponential model fits.
所以如果数据在半对数图上 看起来是直的，指数模型就适用。
Before you go, three things to remember.
结束之前，三个要点。
First, an exponential grows by repeated multiplication, so it eventually beats any linear or polynomial model.
第一，指数以反复相乘的方式增长，所以它终将超过任何线性或多项式模型。
Second, the logarithm is the inverse of the exponential — use it to bring an unknown exponent down and solve.
第二，对数是指数的反函数——用它把未知的指数拿下来求解。
Third, on a semi-log plot an exponential becomes a straight line.
第三，在半对数图上，指数会变成 一条直线。
Keep these in mind, and Unit Two is yours.
记住这些，第二单元就是你的了。