课程： 3. 5 对数函数 | The Way To Learn

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对对数函数

Ears are amazing.
::耳朵是惊人的。

The average human ear can just as easily recognize the whisper of the person in the next chair as the roar of a jet plane on takeoff. Most people know that, but many people do not realize that the difference in power between the two is approximately 10,000,000,000 times!
::普通人的耳朵可以很容易地辨识出在下一椅子上的人的耳语,就像起飞时喷气式飞机的咆哮一样。大多数人知道这一点,但许多人不知道两者之间的权力差异大约是10,000万倍!

If you look up the decibel ratings of these two sounds, you will find that the quietest whispers are apx 10 db, and that a 747 on takeoff can hit 120 db.
::如果你查查这两个声音的分解率, 你会发现最安静的低语是Apx 10 db, 起飞时747分可以击中120 db。

That's only a difference of 100 db, how can that be the same as 10 BILLION times?
::那只是100公分的差数, 怎么可能和10次BILLION一样呢?

Logarithmic Functions
::对对数函数

Every exponential expression can be written in logarithmic form. For example, the equation x = 2 ^y is written as follows: y = log ₂ x . In general, the equation log _b n = a is equivalent to the equation b ^a = n . That is, b is the base , a is the exponent , and n is the power, or the result you obtain by raising b to the power of a. Notice that the exponential form of an expression emphasizes the power, while the logarithmic form emphasizes the exponent. More simply put, a logarithm (or “log” for short) is an exponent.
::每个指数表达式都可以以对数形式写入。例如, 公式 x = 2y 的写法如下: y = log2x。一般来说, 等式logbn = a 等于等式 ba = n 。也就是说, b 是基数, a 是引号, n 是功率, 或通过调出 b 到 a 的功率获得的结果。请注意, 一种表达式的指数形式强调功率, 而对数表则强调 exponent 。更简单地说, logarithm (或简称的“log”) 是 expent 。

Perhaps the most common example of a logarithm is the Richter scale, which measures the magnitude of an earthquake. The magnitude is actually the logarithm base 10 of the amplitude of the quake. That is, m = log ₁₀ A . This means that, for example, an earthquake of magnitude 4 is 10 times as strong as an earthquake with magnitude 3. We can see why this is true of we look at the logarithmic and exponential forms of the expressions: An earthquake of magnitude 3 means 3 = log ₁₀ A . The exponential form of this expression is 10 ³ = A . Thus the amplitude of the quake is 1,000. Similarly, a quake with magnitude 4 has amplitude 10 ⁴ = 10,000.
::也许最常见的对数示例是测算地震规模的Richter尺度。测算地震规模的 Richter 尺度。数值实际上是地震振幅的对数基 10 。也就是说, m = log10A 。这意味着, 例如, 4 级的地震比 3 级的地震强十倍于 3 级的地震。我们可以看到,为什么我们看这些表达式的对数和指数形式是真实的: 3 级的地震 3 表示3 = log10A 。这个表达式的指数形式是 103 = A 。因此, 地震的振幅是 1 000 。同样, 4 级的地震有 104 = 10 000 。

Solving Logarithmic Equations
::解析对数等量

In general, to solve an equation means to find the value(s) of the variable that makes the equation a true statement. To solve log equations, we have to think about what “log” means .
::一般来说, 要解决方程式, 需要找到使方程式成为真实语句的变量值。要解决日志方程式, 我们必须思考“ log” 的含义。

Consider the equation log ₂ x = 5 . What is the exponential form of this equation?
::考虑方程式 log2x = 5。这个方程式的指数形式是什么 ?

The equation log ₂ x = 5 means that 2 ⁵ = x . So the solution to the equation is x = 2 ⁵ = 32.
::方程式 log2x = 5 表示 25 = x 。方程式的解决方案是 x = 25 = 32。

In some log equations, both sides of the equation contain a log. To solve these equations, use the following rule: log _b f ( x ) = log _b g ( x ) → f ( x ) = g ( x ).
::在某些日志方程式中, 公式的两边都包含一个日志。要解析这些方程式, 请使用以下规则 : logbf( x) = logbg( x) = logbg( x) f( x) = g( x) 。

In other words, set the bases equal and solve for the variable in the exponent by treating the exponents on both sides of the equation as simple polynomials.
::换句话说,将方程两侧的推手作为简单的多面体来对待,以此来设定标注变量的等值和求解基数。

Examples
::实例

Example 1
::例1

Rewrite each exponential expression as a log expression.
::将每个指数表达式重写为日志表达式。

In order to rewrite an expression, you must identify its base, its exponent, and its power.
::要重写一个表达式, 您必须确定它的底部、亮点和权力。

3 ⁴ = 81

The 3 is the base, so it is placed as the subscript in the log expression. The 81 is the power, and so it is placed after the “log”. Thus we have: 3 ⁴ = 81 is the same as log ₃ 81 = 4. To read this expression, we say “the logarithm base 3 of 81 equals 4.” This is equivalent to saying “ 3 to the 4 ^th power equals 81.”
::3 是基数, 因此它被放在对数表达式中的下标。 81 是权力, 因此它被放在“ log” 之后。因此, 我们有: 34= 81 是相同的 aslog 381 = 4. 读这个表达式, 我们说“ 81 中的对数基 3 等于 4. ” , 这相当于 “ 3 到 4 ower 等于 81 ” 。

b ^{4

x} = 52
::b4x=52

The b is the base, and the expression 4x is the exponent, so we have: log _b 52 = 4 x . We say, “log base b of 52, equals 4 x .”
::b 是基数, 4x表达式是指数, 所以我们有: logb52 = 4x。我们说, “ log b b of 52, 等于 4x ” 。

Example 2
::例2

Evaluate the function f ( x ) = log ₂ x for the following values.
::为以下值评价函数 f(x) = log2x。

$\begin{aligned} x = 2 \end{aligned}$
::x=2x=2

If $\begin{aligned} x = 2 \end{aligned}$ , we have: f ( x ) = log ₂ x f (2) = log ₂ 2.
::如果 x=2, 我们有: f(x) = log2xf(2) = log22。

To determine the value of log ₂ 2, you can ask yourself: “2 to what power equals 2?” Answering this question is often easy if you consider the exponential form: 2 ^? = 2.
::要确定 log22 的值,你可以问自己:“2 至 2 等于 2 ?” 回答这个问题通常很容易,如果你考虑指数形式: 2?= 2。

The missing exponent is 1. So we have f (2) = log ₂ 2 = 1.
::缺少的指数是 1, 所以我们有f(2) = log22 = 1。

$\begin{aligned} x = 1 \end{aligned}$
::x=1 x=1

If $\begin{aligned} x = 1 \end{aligned}$ , we have: f ( x ) = log ₂ x f (1) = log ₂ 1. As we did in (a), we can consider the exponential form: 2 ^? = 1.
::如果x=1,我们有:f(x) = log2xf(1) = log21。正如我们在(a) 中所做的那样,我们可以考虑指数形式: 2?= 1。

The missing exponent is 0. So we have f (1) = log ² 1 = 0.
::缺少的指数是 0。所以我们有f(1) = log21 = 0。

$\begin{aligned} x = - 2 \end{aligned}$
::x% 2

If $\begin{aligned} x = - 2 \end{aligned}$ , we have: f ( x ) =log ₂ x f (-2) = log ₂ -2. Again, consider the exponential form: 2 ^? = -2.
::Ifx=% 2, 我们有: f( x) =log2xf( 2) = log2xf( 2) = log2- 2。请再次考虑指数表 : 2? = - 2 。

There is no such exponent. Therefore f (-2) = log ₂ -2 does not exist .
::没有这样的指数。因此f( 2) = log2-2 不存在。

Example 3
::例3

Solve the equation log ₂ (3 x -1) = log ₂ (5 x - 7) for x.
::解决x的方程式对数对数2( 3x-1) = log2( 5x - 7) 。

Because the logarithms have the same base (2), the arguments of the log (the expressions 3 x - 1 and 5 x - 7) must be equal . So we can solve as follows:
::因为对数有相同的基数(2), 日志的参数( 表达式 3x-1 和 5x-7) 必须相等。所以我们可以解析如下 :

		log ₂ (3 x -1)	= log ₂ (5 x - 7)
		3 x - 1	= 5 x - 7
		+7 \|+7
		3 x + 6	= 5 x
		-3 x
		6	= 2 x
		x	= 3

Example 4
::例4

Rewrite the following logarithmic expressions in exponential form.
::以指数形式重写以下对数表达式。

log ₁₀ 100 = 2
::log10100 = 2

The base is 10, and the exponent is 2, so we have 10 ² = 100.
::基数是10, 指数是2, 所以我们有102=100。

log _b w = 5
::logbw = 5

The base is b, and the exponent is 5, so we have b ⁵ = w .
::基数是b, 指数是5, 所以我们有b5=w。

Example 5
::例5

Solve each equation for $\begin{aligned} x \end{aligned}$ .
::为 x 解决每个方程式。

log ₄ x = 3
::log4 x = 3

Writing the equation in exponential form gives us the solution: x = 4 ³ = 64.
::以指数形式写出方程式, 给我们提供了答案: x = 43 = 64 。

log ₅ (x + 1) = 2
::log5(xx+1) = 2

Writing the equation in exponential form gives us a new equation: 5 ² = x + 1.
::以指数形式写入方程式时, 给我们提供了一个新的方程式: 52 = x + 1 。

We can solve this equation for x :
::我们可以解析 x 的这个方程式 :

		5 ²	= x + 1
		25	= x + 1
		x	= 24

1 + 2log ₃ (x - 5) = 7
::1 + 2log3(x-5) = 7

First, we have to isolate the log expression:
::首先,我们必须分离日志表达式:

		1 + 2log ₃ ( x - 5)	= 7
		2log ₃ ( x - 5)	= 6
		log ₃ (x - 5)	= 3

Now, we can solve the equation by rewriting it in exponential form:
::现在,我们可以通过重写指数形式来解决方程式问题:

		log ₃ (x - 5)	= 3
		3 ³	= x - 5
		27	= x - 5
		x	= 32

Review
::回顾

State the expression in English: $\begin{aligned} l o g_{3} 243 \end{aligned}$ .
::英文表示: log3243。

Write the given equation in logarithmic form: $\begin{aligned} (\frac{1}{3})^{- 5} = 243 \end{aligned}$ .
::以对数表写入给定方程式: (13)- 5=243。

Determine if the two equations below are equivalent and label them according to function family: $\begin{aligned} y = l o g_{a} x \end{aligned}$ and $\begin{aligned} x = a^{y} \end{aligned}$ .
::确定以下两个方程式是否相等, 并按函数家族 y=logax 和 x=ay 标记它们。

Evaluate the logarithms:
::评估对数 :

$\begin{aligned} l o g_{4} 625 \end{aligned}$
::log4625

$\begin{aligned} l o g_{6} 64 \end{aligned}$
::log664 log664

$\begin{aligned} l o g_{3} 216 \end{aligned}$
::log3216

Evaluate each logarithm at the indicated value of x.
::按 x 的表示值评价每项对数。

$\begin{aligned} f (x) = l o g_{2} x \end{aligned}$ for $\begin{aligned} x = 32 \end{aligned}$
:fx) = x= 32 的log2x

$\begin{aligned} f (x) = l o g_{3} x \end{aligned}$ for $\begin{aligned} x = 1 \end{aligned}$
: fx) = x=1 的log3x

$\begin{aligned} f (x) = l o g_{4} x \end{aligned}$ for $\begin{aligned} x = 2 \end{aligned}$
:xx) = x=2 的log4x

$\begin{aligned} f (x) = l o g_{10} x \end{aligned}$ for $\begin{aligned} x = \frac{1}{100} \end{aligned}$
:fx) = log10x x= 1100

Solve for x :
::解决 x:

$\begin{aligned} l o g_{3} (2 x + 2) = l o g_{3} (x - 4) \end{aligned}$
::对数 3 (2x+2) = log3 (x- 4)

$\begin{aligned} l o g_{7} (\frac{x}{3}) = l o g_{7} (x - 4) \end{aligned}$
::log7(x3) =log7(x- 4)

$\begin{aligned} l o g_{4} \sqrt[3]{8 x} = l o g_{4} 8 \end{aligned}$
::log48x3 = log48

$\begin{aligned} l o g_{5} (x + 9) = l o g_{5} 3 (x - 2) \end{aligned}$
::log5(x+9) = log53(x-2)

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

常规

对对数函数

Logarithmic Functions ::对对数函数

Solving Logarithmic Equations ::解析对数等量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源