Course: 3.7 对数属性 | The Way To Learn

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对数属性的对数属性

What is the value of the expression $\begin{aligned} l o g_{6} (8) + l o g_{6} (27) \end{aligned}$ ?
::表达式对数 6(8)+log6(27)的值是多少?

Alone, neither of these expressions has an integer value, therefore combining them might seem like a bit of a challenge. The value of log ₆ 8 is between 1 and 2; the value of log ₆ 27 is also between 1 and 2.
::单靠这些表达式,这两个表达式都没有整数值, 因此合并起来可能有点困难。 log6 8 的值介于 1 和 2 之间; log6 27 的值介于 1 和 2 之间。

Is there an easier way?
::有更简单的方法吗?

Properties of Logarithms
::对数属性的对数属性

Previously we defined the logarithmic function as the inverse of an exponential function, and we evaluated log expressions in order to identify values of these functions. In this lesson we will work with more complicated log expressions. We will use the properties of logarithms to write a log expression as the sum or difference of several expressions, or to write several expressions as a single log expression.
::以前,我们定义对数函数为指数函数的反值,我们评估了日志表达式,以辨别这些函数的值。在这个课程中,我们将使用更复杂的日志表达式。我们将使用对数的属性来将日志表达式作为数个表达式的总和或差数写入,或者将数个表达式作为单日志表达式写入。

Properties of Logarithms
::对数属性的对数属性

Because a logarithm is an exponent, the properties of logs reflect the properties of exponents.
::由于对数是指数,日志的属性反映了指数的属性。

The basic properties are:
::基本财产是:

$\begin{aligned} l o g_{b} (x y) = l o g_{b} x + l o g_{b} y \end{aligned}$
::logb(xy) =logbx+logby(xy) =logbx+logby

$\begin{aligned} l o g_{b} (\frac{x}{y}) = l o g_{b} x - l o g_{b} y \end{aligned}$
::logb(xy) =logbx-logby

$\begin{aligned} l o g_{b} x^{n} = n l o g_{b} x \end{aligned}$
::logbxn = logbx

Expanding Expressions
::扩展表达式

Using the properties of logs, we can write a log expression as the sum or difference of simpler expressions. Consider the following examples:
::使用日志的属性,我们可以将日志表达式作为简单表达式的总和或差数写入日志表达式。

$\begin{aligned} l o g_{2} 8 x \end{aligned}$ = $\begin{aligned} l o g_{2} 8 + l o g_{2} x \end{aligned}$ = $\begin{aligned} 3 + l o g_{2} x \end{aligned}$
::log28x = log28+log2x = 3+log2x = 3+log2x

$\begin{aligned} l o g_{3} (\frac{x^{2}}{3}) \end{aligned}$ = $\begin{aligned} l o g_{3} x^{2} - l o g_{3} 3 \end{aligned}$ = $\begin{aligned} 2 l o g_{3} x - 1 \end{aligned}$
::log3 (x23) = log3x2 - log33 = 2log3x- 1

Using the log properties in this way is often referred to as "expanding". In the first example, expanding the log allowed us to simplify, as log ₂ 8 = 3. Similarly, in the second example, we simplified using the log properties, and the fact that log ₃ 3 = 1.
::以这种方式使用日志属性通常被称为“扩展 ” 。在第一个例子中,扩大日志使我们得以简化,如 log2 8 = 3. 同样,在第二个例子中,我们简化了对日志属性的使用,以及对日志3 3 = 1 这一事实。

Examples
::实例

Example 1
::例1

To condense a log expression, we will use the same properties we used to expand expressions. Consider the expression $\begin{aligned} l o g_{6} (8) + l o g_{6} (27) \end{aligned}$ . Individually, neither of these expressions has an integer value. The value of log ₆ 8 is between 1 and 2; the value of log ₆ 27 is also between 1 and 2.
::要压缩日志表达式, 我们将使用用于扩展表达式的相同属性。请考虑表达式 log6( 8)+log6( 27) 。单独地说, 这些表达式都没有整数值。 log6 8 的值介于 1 和 2 之间; log6 27 的值介于 1 和 2 之间。

However, if we condense the expression, we get:
::然而,如果我们压缩表达方式,我们就会得到:

$\begin{aligned} l o g_{6} (8) + l o g_{6} (27) = l o g_{6} (8 \cdot 27) = l o g_{6} (216) = 3 \end{aligned}$
::log6(8)+log6(27)=log6(8)27=log6(216)=3

Example 2
::例2

Expand each expression.
::放大每个表达式。

$\begin{aligned} l o g_{5} 25 x^{2} y \end{aligned}$
::log525x2y

$\begin{aligned} l o g_{5} (25) x^{2} y = l o g_{5} (25) + l o g_{5} x^{2} + l o g_{5} y = 2 + 2 l o g_{5} x + l o g_{5} y \end{aligned}$
::log5 (25)x2y=log5 (25)+log5x2+log5y=2+2log5xx+log5y

$\begin{aligned} l o g_{10} (\frac{100 x}{9 b}) \end{aligned}$
::对数对数 10( 100x9b)

$\begin{aligned} l o g_{10} (\frac{100 x}{9 b}) \end{aligned}$	$\begin{aligned} = l o g_{10} 100 x - l o g_{10} 9 b \end{aligned}$
	$\begin{aligned} = l o g_{10} 100 + l o g_{10} x - [l o g_{10} 9 + l o g_{10} b] \end{aligned}$
	$\begin{aligned} = 2 + l o g_{10} x - l o g_{10} 9 - l o g_{10} b \end{aligned}$

Example 3
::例3

Condense the expression 2log ₃ x + log ₃ 5 x - log ₃ ( x + 1).
::折合表达式 2log3x + log3 5x - log3 (x + 1) 。

$\begin{aligned} 2 l o g_{3} x + l o g_{3} 5 x - l o g_{3} (x + 1) = l o g_{3} x_{2} + l o g_{3} 5 x - l o g_{3} (x + 1) \end{aligned}$
::2log3x+log35x-log3(x+1)=log3x2+log35x-log3(x+1)

$\begin{aligned} = l o g_{3} (x^{2} (5 x)) - l o g_{3} (x + 1) \end{aligned}$
::=log3(x2(5x))-log3(x+1)

$\begin{aligned} = l o g_{3} (\frac{5 x^{3}}{x + 1}) \end{aligned}$
::=log3 (5x3x+1)

Note that not all solutions may be valid, since the argument must be defined. For example, the expression above: $\begin{aligned} (\frac{5 x^{3}}{x + 1}) \end{aligned}$ is undefined if x = -1.
::请注意,并非所有解决方案都可能有效, 因为参数必须定义。例如, 上面的表达式: (5x3x+1) 如果 x = - 1, 则未定义。

Example 4
::例4

Condense the expression log ₂ (x ² - 4) - log ₂ ( x + 2).
::吸收表达式对数( x2 - 4) - log2 (x + 2) 。

$\begin{aligned} l o g_{2} (x^{2} - 4) - l o g_{2} (x + 2) = l o g_{2} (\frac{x^{2} - 4}{x + 2}) \end{aligned}$
::log2 (x2) - 4) - log2 (x2+2) = log2 (x2) - 4x+2)

$\begin{aligned} = l o g_{2} (\frac{(x + 2) (x - 2)}{x + 2}) \end{aligned}$
::=log2(x+2)(x-2)x+2)

$\begin{aligned} = l o g_{2} (x - 2) \end{aligned}$
::=log2(x-2) =log2(x-2)

Note that the argument of a log must be positive. For example, the expressions above are not defined for x ≤ 2 (which allows us to "cancel" ( x +2) without worrying about the condition x ≠ -2).
::请注意日志的参数必须是正数。例如, 上面的表达式没有被定义为 x 2 (它允许我们“ 取消 ” (x+ 2) 而不担心条件 x - 2 ) 。

Example 5
::例5

Condense the following into a single logarithm: $\begin{aligned} 3 l o g_{6} x + 2 l o g_{6} (3 x) - l o g_{6} (2 x^{3}) \end{aligned}$ .
::将以下内容合并成单对数: 3log6x+2log6(3x)- log6(2x3) 。

Recall that $\begin{aligned} 3 l o g_{x} y = l o g_{x} y^{3} \end{aligned}$
::回想 3 logxy=logxy3 时

$\begin{aligned} 3 l o g_{6} x + 2 l o g_{6} (3 x) - l o g_{6} (2 x^{3}) \to \end{aligned}$
::3log6x+2log6( 3x) - log6( 2x3)

$\begin{aligned} l o g_{6} (x^{3} + 3 x^{2}) - l o g_{6} (2 x^{3}) \to \end{aligned}$
::log6(x3+3x2) -log6(2x3)

$\begin{aligned} l o g_{6} (\frac{x^{3} + 3 x^{2}}{2 x^{3}}) \to \end{aligned}$
::对数 6( x3+3x22x3) {}

$\begin{aligned} l o g_{6} (\frac{x + 3}{2 x}) \end{aligned}$
::对数 6(x+32x)

Example 6
::例6

Expand the logarithm: $\begin{aligned} l o g_{2} (\frac{5 x^{7}}{3 x^{4}}) \end{aligned}$ .
::展开对数: log2( 5x73x4)。

Reversing a previously used rule gives $\begin{aligned} l o g_{x} (\frac{y}{z}) = l o g_{x} y - l o g_{x} z \end{aligned}$ .
::更改先前使用的规则时, 给予对数( yz) = logxy- logxz 。

$\begin{aligned} l o g_{2} (\frac{5 x^{7}}{3 x^{4}}) \to \end{aligned}$
::log2( 5x73x4)____________________________________________________

$\begin{aligned} l o g_{2} (\frac{5 x^{3}}{3}) \to \end{aligned}$ (reducing the fraction first)
::log2(5x33)}(先降低分数)

$\begin{aligned} l o g_{2} 5 x^{3} - l o g_{2} 3 \end{aligned}$
::log23 - log23

Review
::回顾

Expand each logarithmic expression:
::展开每个对数表达式 :

$\begin{aligned} l o g_{5} (a b) \end{aligned}$
::log5(ab) 日志5(ab)

$\begin{aligned} l o g_{6} \frac{a}{\sqrt{3} b} \end{aligned}$
::对数 6a3b

$\begin{aligned} l o g_{6} \frac{a b}{c} \end{aligned}$
::对数 6abc

If $\begin{aligned} v = l o g_{x} (\frac{4 z^{2}}{y^{3}}) \end{aligned}$ expand $\begin{aligned} v \end{aligned}$
::如果 v=logx( 4z2y3) 扩展 v

$\begin{aligned} l o g_{2} (\frac{4 x^{3}}{\sqrt{y}}) \end{aligned}$
::log2( 4x3y)

If $\begin{aligned} R = l o g_{3} (\frac{2 G M}{c^{2}}) \end{aligned}$ expand $\begin{aligned} R \end{aligned}$
::如果 R=log3( 2GMc2) 扩展 R

Condense each logarithmic expression:
::集中每个对数表达式 :

$\begin{aligned} l o g_{5} A + l o g_{5} C \end{aligned}$
::对数 5A+log5C

$\begin{aligned} \frac{1}{2} l o g_{2} C - l o g_{2} B \end{aligned}$
::12log2C-log2B 12log2C-log2B

$\begin{aligned} 2 l o g_{b} x + 2 l o g_{b} y \end{aligned}$
::2logbx+2logby 2logbx+2logby 2logbx+2logby 2logbx+2logby 2logbx+2logbex+2logblogby 2logbx+2logbx+2logby 2logbx+2logblogby 2logbx+2logbx+2logblogby 2logbx+2 logblogbx+2 logblogby 2logbx+2 logblogb x 2logbx+2 logbx+2 logblogbx+2 logblogbx+2 logb2 logb x 2logbx+2 logby 2logb2 logb2 logb2 logb2 logb x 2logb x 2logbx+2 logb x 2 logb2 logb2 logb x 2logbx+2 logbx+2 logb2 logb2 logbx+2 logb x 2 logb2 logb2 logb2 logb2 2 log2 log2 log2 2 log2 2 2 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 以 2 log x 2 log x 2 x 2 x 2 x+ 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x+ 2 log x+2 x 2 x 2 x+2 logb x+2 logb x+2 logb x+2 x+ 2 logb x+ 2 以 2 以 2 以 2 以 2 以 2 logb x 2 以 2 以 2 以 2 x x+2 以 log x 以以以

$\begin{aligned} 6 l o g_{10} a + l o g_{10} b \end{aligned}$
::6log10a+log10b 6log10a+log10b

$\begin{aligned} 2 l o g_{3} a + 4 l o g_{3} b - l o g_{3} c \end{aligned}$
::2log3a+4log3b-log3c

$\begin{aligned} \frac{1}{2} l o g_{4} w - 5 l o g_{4} z \end{aligned}$
::12log4w-5log4z 12log4w-5log4z

$\begin{aligned} (l o g_{10} x + l o g_{10} y) - l o g_{10} w \end{aligned}$
:log10x+log10y) - log10w

Simplify:
::简化 :

$\begin{aligned} l o g_{10} A^{3} - l o g_{10} B^{\frac{2}{3}} + l o g_{10} A^{\frac{1}{3}} + l o g_{10} B^{\frac{5}{3}} \end{aligned}$
::log10A3 - log10B23+log10A13+log10B53

$\begin{aligned} 2 l n (A B) - l n (\frac{B}{A}) \end{aligned}$
::2(AB)-(BA)

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。

Section outline

General

对数属性的对数属性

Properties of Logarithms ::对数属性的对数属性

Properties of Logarithms ::对数属性的对数属性

Expanding Expressions ::扩展表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Properties of Logarithms
::对数属性的对数属性

Properties of Logarithms
::对数属性的对数属性

Expanding Expressions
::扩展表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)