11.10 解决指数等号

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  解决指数等号

  Some businesses choose to depreciate their equipment exponentially. Let's say the photocopier below originally sold for $3,100, and it loses 20% of its value each year. We can model that with a function in the form  $\begin{array}{r}V=3100(0.8{)}^t,\end{array}$  where  $\begin{array}{r}V\end{array}$   is the value of the photocopier in dollars, and  $\begin{array}{r}t\end{array}$  is time in years. How long will it take for the value of the photocopier to fall below $500? 

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  To answer this question, we need to be able to solve an exponential equation . We learn how to do that in this section.  
  ::为了回答这个问题,我们需要能够解决一个指数式的方程式。我们在本节学习如何做到这一点。

  

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  Solving Exponential Equations
  ::解决指数等号

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  It is helpful to classify exponential equations in two categories—those where we can express both sides with the same base, and those where we cannot. If we can express both sides with the same base, that will likely require less work. 
  ::将指数式方程式分为两类是有益的 — — 一类是我们可以以同样的基数表达双方,另一类是无法表达的。 如果我们能以同样的基数表达双方,那可能需要更少的工作。

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      if Both Sides Can Be Expressed With the Same Base
  ::如果两边都能用同一基地表达的话

  If  $\begin{array}{r}{b}^u=b^v\end{array}$ , then  $\begin{array}{r}u=v\end{array}$ .  

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  Example 1
  ::例1

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  Solve  $\begin{array}{r}{8}^x=128\end{array}$ . 
  ::解决8x128。

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  Solution:  If we express 8 and 128 as  powers of 2, we can set the exponents equal to each other and solve.
  ::解决方案:如果我们把8和128表达为2的权力, 我们可以把推论者放在对等的位置,解决。

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  $$\begin{array}{rl}{8}^x& =128\\ 2^{3x}& =2^7\\ 3x& =7\\ x& =\frac{7}{3}\end{array}$$

  
  ::8x=12823x=273x=7x=73

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  So, $\begin{array}{r}{8}^{\frac{7}{3}}=128\end{array}$ .
  ::所以,873=128。

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  Example 2
  ::例2

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  Solve   $\begin{array}{r}{4}^{x-8}=16\end{array}$ .
  ::解决4x8=16。

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  Solution: Change 16 to $\begin{array}{r}{4}^2\end{array}$ and set the exponents equal to each other.
  ::解决办法:将16修改为42,将引号对等。

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  $$\begin{array}{rl}{4}^{x-8}& =16\\ 4^{x-8}& =4^2\\ x-8& =2\\ x& =10\end{array}$$

  
  ::4x-8=164x-8=42x-8=2x=10

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  Example 3
  ::例3

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  Solve $\begin{array}{r}{10}^{x-3}={100}^{3x+11}\end{array}$ .
  ::解决 10x-3 = 1003x+11 。

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  Solution: Change 100 into a power of 10.
  ::解决办法:将100变成10的功率。

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  $$\begin{array}{rl}{10}^{x-3}& ={10}^{2(3x+11)}\\ x-3& =6x+22\\ \text{-}25& =5x\\ \text{-}5& =x\end{array}$$

  
  ::10x-3=102(3x+11)x-3=6x+22-25=5x-5=x

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  by Mathispower4u demonstrates how to solve exponential equations when both sides can be written with the same base. 
  ::由 Mathispower4u 演示如何解析指数方程式, 当两边可以用相同的基数写入时 。

   

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  Example 4
  ::例4

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  Solve  $\begin{array}{r}{e}^{2x}=\frac{1}{e^{4x-6}}\end{array}$ . 
  ::解决 e2x=1e4x-6。

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  Solution: 
  ::解决方案 :

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  $$\begin{array}{rl}{e}^{2x}& =\frac{1}{e^{4x-6}}\\ e^{2x}& =e^{\text{-}(4x-6)}\\ e^{2x}& =e^{\text{-}4x+6}\\ \\ 2x& =\text{-}4x+6\\ 6x& =6\\ x& =1\end{array}$$

  
  ::e2x=1e4x-6e2x=e-(4x-6)e2x=e-4x+62x=-4x+66x=6x=1

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  But what happens when a common base is not easily found? We must use logarithms and the power rule.
  ::但是,当共同基础不易找到时会怎样呢? 我们必须使用对数和权力规则。

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    Solving Exponential Equations if Both Sides Cannot Be Expressed With the Same Base
  ::如果两面无法以同一基点表达,则解决指数等号

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  Steps: 
  ::步骤:

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  1. Isolate the term with the variable in the exponent on one side.
  ::1. 将术语与表列中的变量分隔开来。

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  2. Take any logarithm of both sides. 
  ::2. 采用双方的任何对数。

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  3. Use the power rule of logarithms to move the exponent from inside the logarithm to outside the logarithm. 
  ::3. 使用对数对数的对数规则将对数从对数内移到对数外。

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  4. Solve. 
  ::4. 解决。

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  Example 5
  ::例5

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  Solve $\begin{array}{r}{6}^x=49\end{array}$ . Round your answer to the nearest three decimal places.
  ::解决 6x=49。 将您的回答转至最接近小数点后三位位数 。

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  Solution: To solve this exponential equation, as the term with the exponent is already isolated, let's take the logarithm of both sides. The most convenient  logs to use are either $\begin{array}{r}\ln \end{array}$ (the natural log) or log (log base 10), since they can be calculated on most calculators. Here, we will use the natural log.
  ::解答: 要解析这个指数式方程式, 因为引数的术语已经孤立, 我们选用两边的对数。 最方便使用的日志要么在( 自然日志) 中, 要么在( log basic 10) 中, 因为它们可以用大多数计算器来计算 。 在这里, 我们将使用自然日志 。

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  $$\begin{array}{rl}{6}^x& =49\\ \ln 6^x& =\ln 49\\ x\ln 6& =\ln 49\\ x& =\frac{\ln 49}{\ln 6}\approx 2.172\end{array}$$

  
  ::6x=49ln_6x=ln_49xln_9x_6=ln_49x=ln_49_6__6}#2.172

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  Note we could have also taken the logarithm base 6 of both sides. 
  ::请注意,我们本来还可以采用双方对数基6。

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  $$\begin{array}{rl}{6}^x& =49\\ {\log }_6{6}^x& ={\log }_{6}49\\ x{\log }_{6}6& ={\log }_{6}49& & \text{identity property of logs}\\ x& ={\log }_{6}49\\ & =\frac{\log 49}{\log 6}\approx 2.172& & \text{change of base formula for calculator}\end{array}$$

  
  ::6x=49log6\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\L\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\L\\\\\\\\\\\\\L\\\\\\\\\\\\\\\\\\\L\\\\\\\\\\\\\\\\\\\\\\\\\\\在计算计算基公式的公式的基公式

   

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  Example 6
  ::例6

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  Solve $\begin{array}{r}{8}^{2x-3}-4=5\end{array}$ .
  ::解决82x-3-4=5。

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  Solution: Add 4 to both sides to isolate the term with the exponent, and then take the log of both sides. Here, we use the common log. Any logarithm is acceptable. 
  ::解决方案 : 向 双方 添加 4 个 , 将 词汇 与 指数 分隔开来 , 然后取 双方 的 日志 。 在这里, 我们使用 共同 日志 。 任何对数都是可以接受的 。

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  $$\begin{array}{rl}{8}^{2x-3}-4& =5\\ 8^{2x-3}& =9\\ \log 8^{2x-3}& =\log 9\\ (2x-3)\log 8& =\log 9\\ 2x-3& =\frac{\log 9}{\log 8}\\ 2x& =3+\frac{\log 9}{\log 8}\\ x& =\frac{3}{2}+\frac{\log 9}{2\log 8}\approx 2.028\end{array}$$

  
  ::82 - 3 - 4= 582x- 3= 9log_ 82x- 3= log_ 9( 2x-3) log_ 8= log_ 92x- 3= log_ 82x= 3+ log_ 9log_ 8x= 3+ log_ 9log_ 8x= 32+log_ 92log_ 8_ @ 2. 028

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  Notice that we did not find the numeric value of $\begin{array}{r}\log 9\end{array}$ or $\begin{array}{r}\log 8\end{array}$ until the very end. This will give us  a more accurate answer because we did not round these values earlier in the problem. 
  ::注意我们直到最后才发现对数=%9 或对数=8 的数值。 这将给我们一个更准确的答案, 因为我们在问题之前没有对这些数值进行循环 。

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  Example 7
  ::例7

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  Solve  $\begin{array}{r}2(7{)}^{3x+1}=48\end{array}$ .
  ::解决 2(7)3x+1=48。

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  Solution:  Divide both sides by 2 and then take the log of both sides.
  ::解决方案:将双方除以2,然后取出双方的日志。

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  $$\begin{array}{rl}2(7{)}^{3x+1}& =48\\ 7^{3x+1}& =24\\ \ln 7^{3x+1}& =\ln 24\\ (3x+1)\ln 7& =\ln 24\\ 3x+1& =\frac{\ln 24}{\ln 7}\\ 3x& =\text{-}1+\frac{\ln 24}{\ln 7}\\ x& =\text{-}\frac{1}{3}+\frac{\ln 24}{3\ln 7}\approx 0.211\end{array}$$

  
  ::2(7)3x+1=4873x+1=24ln73x+1=ln24(3x+1) =7=ln 2443x+1=ln 2443x+1=_24x=-1+0}7x=-1+ln_24ln_24ln_7x=-13+ln_243ln}_7_0.211

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  by Mathispower4u shows how to solve exponential equations when it is difficult to write both sides with the same base. 
  ::使用 Mathispower4u 来显示当难以用相同的基数书写两边时, 如何解析指数方程式 。

   

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  Example 8
  ::例8

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  Solve  $\begin{array}{r}\frac{2}{3}\cdot 5^{x+2}+9=21\end{array}$ .
  ::解决 235x+2+9=21。

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  Solution: Subtract 9 from both sides and multiply both sides by $\begin{array}{r}\frac{3}{2}.\end{array}$  Then take the log of both sides.
  ::解决方案:从两侧减9, 乘以两侧32, 然后取出两侧的日志。

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  $$\begin{array}{rl}\frac{2}{3}\cdot 5^{x+2}+9& =21\\ \frac{2}{3}\cdot 5^{x+2}& =12\\ 5^{x+2}& =18\\ (x+2)\log 5& =\log 18\\ x& =\frac{\log 18}{\log 5}-2\approx \text{-}0.204\end{array}$$

  
  ::23_5x+2+9=2123_5x+2=125x+2=18(x+2),log_5=log_18x=log_18log_5-2_0.204

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  Example 9
  ::例9

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  A  photocopier originally sold for $3,100, and it loses 20% of its value each year. We can model that with a function in the form $\begin{array}{r}V=3100(0.8{)}^t,\end{array}$  where $\begin{array}{r}V\end{array}$  is the value of the photocopier in dollars, and $\begin{array}{r}t\end{array}$   is time in years. How long will it take for the value of the photocopier to fall below $500? 
  ::一台复印机最初售价为3,100美元,每年损失其价值的20%。 我们可以以V=3100(0.8)t(V=3100(0.8)t,V是复印机的美元价值)和T(年年时间)等函数来模拟它。 复印机的价值要跌到500美元以下还要花多久?

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  Solution:  We isolate the term with the exponent and then take the log of both sides. 
  ::解决:我们把这个词与推手隔离开来,然后记录双方的情况。

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  $$\begin{array}{rl}500& =3,100(0.8{)}^t\\ \frac{500}{3,100}& =(0.8{)}^t\\ \log (\frac{5}{31})& =\log (0.8{)}^t\\ \log (\frac{5}{31})& =t\log (0.8)\\ t& =\frac{\log (\frac{5}{31})}{\log (0.8)}\approx 8.2\text{years}\end{array}$$

  
  ::500=3100(0.8)t 5003100=(0.8)tlog(531)=(0.8)tlog(0.8)tlog(531)=(0.8)tlog(0.8)t=(531)log(0.8)_(8.2)

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  It will take about 8.2 years to depreciate to $500, or about 8 years and 2 1/2 months.  
  ::折旧到500美元大约需要8.2年,大约8年零2个月半。

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     How To Solve Exponential Equations With Desmos
  ::如何用消逝来解脱指数等同

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  As with other equations, we can find the solution to an exponential equation by graphing each side of the equation and finding the point of intersection . To get to the exponent, use the caret key, ^ ( Shift 6). For example, to find the solution of  $\begin{array}{r}{3}^{x+2}=4\end{array}$ , we graph each side of the equation and find the point of intersection.  
  ::与其它方程式一样,我们可以找到指数方程的解决方案,方法是绘制方程的每个侧面图和找到交叉点。要到达前方,请使用关节键 , (Shift 6) 。 例如,要找到 3x+2= 4 的解决方案, 我们绘制方程的每个侧面图并找到交叉点 。

  

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  The solution is -0.738.
  ::解决办法是 -0.738。

   

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     How To Solve Exponential Equations With a TI-83/84
  ::如何用TI-83/84 解决指数等同

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  As with other equations, we can find the solution to an exponential equation by graphing each side of the equation and finding the point of intersection. To get to the exponent, press  the caret key, ^ . For example, to find the solution of  $\begin{array}{r}{3}^{x+2}=4\end{array}$ , we graph each side of the equation and find the point of intersection. To find the point of intersection, press 2 ^nd , TRACE , and then select Intersect. You will then have to press ENTER three times to find the intersection point. 
  ::与其它方程式一样, 我们可以找到指数式方程式的解决方案, 方法是绘制方程式的每个侧面图并找到交叉点。 要到达提示点, 请按 {} 。 例如, 要找到 3x+2=4 的解决方案, 我们就可以按 3x+2=4, 方程式的每个侧面图并找到交叉点 。 要找到交叉点, 请按 2nd, TRACE , 然后选择交叉点 。 然后您需要按 ENTER 三次才能找到交叉点 。

  

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  The solution is -0.738.
  ::解决办法是 -0.738。

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  Feature: Moore, Better, Faster
  ::特写:摩尔、更好、更快

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  by Deirdre Mundy
  ::由Deirdre Mundy 编辑

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  The first computers took up entire rooms. They spent hours doing simple calculations. Yet they had less computing power than the typical smartphone has today. In less than 50 years, we have created devices that make the science fiction of the past look shortsighted. How did computers change so quickly?
  ::第一台计算机占用了整个房间,他们花了几个小时做简单的计算。然而,他们现在的计算能力比典型的智能手机要低。 在不到50年的时间里,我们创造了一些设备,使过去的科幻小说看起来短视。 计算机是如何如此迅速地变化的?

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  In the 1960s, an engineer named Gordon Moore noticed that microchips were improving at a phenomenal rate . He predicted that the number of transistors that scientists could fit on a silicon chip would double every two years—an exponential function! By fitting more processing power onto each chip, engineers could create faster, smaller, more versatile computers every two years. Moore's law predicted that these increases would continue at an exponential rate. As technology has improved, computers have become not only faster but also less expensive. In 1977, an Apple II computer cost $2,638. It had 48 kilobytes (kB) of RAM, and its processor ran at speeds of 1 Megahertz (MHz). Today, the iPhone 5 sells for $150 with a cell phone plan. It has 1,016 megabytes (MB) of RAM and a processor that runs at 1300 MHz, which makes it 1,300 times faster than the early Apple computers and capable of storing more than 20,000 times the memory. It also costs less than 6% of the original price, even if inflation is ignored. You have more computing power in your mobile device than scientists of the '80s had in their labs.
  ::1960年代,一位名叫Gordon Moore的工程师注意到微小芯片正在以惊人的速度改善。他预测科学家可以安装在硅芯上的晶体管的数量每两年会翻一番——一个指数函数!通过在每个芯片上安装更多处理器,工程师可以每两年制造更快、更小、更多的计算机。摩尔的法律预测这些增长将以指数速率继续。随着技术的改进,计算机不仅速度更快,而且成本也更低。1977年,一台苹果II计算机成本为2,638美元。它有48千字节RAM,其处理器速度为1兆赫。今天,iPhone 5以手机计划的价格销售150美元。它拥有1,016兆字节的RAM和1300兆赫的处理器,这使得它比早期的苹果计算机更快1,300倍,并且能够存储超过20,000倍的记忆。它的成本也低于原价的6 %,即使通货膨胀被忽略了。在80年代的实验室里,你的移动设备的计算能力比科学家还要多。

  

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  Engineers predict that we will soon reach the limits of Moore's law. Right now, engineers can create transistors that measure 14-22 nanometers (nm) across. By 2020, they will have reached the 5-7 nm range . After that, it will likely become too expensive to develop smaller transistors, and the exponential increases will stop.
  ::工程师们预测我们很快就会达到摩尔法律的极限。 现在,工程师们可以创建测量14-22纳米的晶体管。 到2020年,他们将达到5-7纳米的射程。 之后,开发较小的晶体管可能变得太贵了,而指数增长将停止。

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    by IBM discusses the extremes of Moore's Law. 
  ::IBM讨论摩尔法律的极端。

   

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  Summary
  ::摘要

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  • To solve exponential equations where each side can be expressed with the same base, write each side with the same base, and then set the exponents equal to each other. 
    ::要解开指数方程式, 每一方方程式可以用相同的基数表达, 以相同的基数写入每方方方程式, 然后将指数方程式设置为等值 。
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  • To solve exponential equations where each side cannot be expressed with the same base, isolate the term with the variable in the exponent, take any logarithm of both sides of the equation, and then use the power rule of logarithms. 
    ::要解析每边无法以相同基数表达的指数方程式, 将术语与引号中的变量分隔开来, 取出方程式两侧的对数, 然后使用对数的功率规则 。

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  Review
  ::回顾

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  Use logarithms and a calculator to solve the equations below for $\begin{array}{r}x\end{array}$ . Round answers to three decimal places.
  ::使用对数和计算器解析 x. x 的以下方程式。 对小数点后三个位数的回合答案 。

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  1.  $\begin{array}{r}{5}^x=65\end{array}$
  ::1. 5x=65

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  2.  $\begin{array}{r}{7}^x=75\end{array}$
  ::2. 7x=75

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  3.  $\begin{array}{r}{2}^x=90\end{array}$
  ::3. 2x=90

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  4.  $\begin{array}{r}{3}^{x-2}=43\end{array}$
  ::4. 3x-2=43

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  5.  $\begin{array}{r}{6}^{x+1}+3=13\end{array}$
  ::5. 6x+1+3=13

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  6.  $\begin{array}{r}6({11}^{3x-2})=216\end{array}$
  ::6. 6(113x-2)=216

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  7.  $\begin{array}{r}8+{13}^{2x-5}=35\end{array}$
  ::7. 8+132x-5=35

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  8.  $\begin{array}{r}\frac{1}{2}\cdot 7^{x-3}-5=14\end{array}$
  ::8. 12-7x-3-5=14

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  Solve the following exponential equations without a calculator.
  ::在没有计算器的情况下解决下列指数方程式。

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  9.  $\begin{array}{r}{4}^x=8\end{array}$
  ::9. 4x=8

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  10.  $\begin{array}{r}{9}^{x-2}=27\end{array}$
  ::10. 9x-2=27

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  11.  $\begin{array}{r}{5}^{2x+1}=125\end{array}$
  ::11. 52x+1=125

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  12.  $\begin{array}{r}{9}^3=3^{4x-6}\end{array}$
  ::12. 93=34x-6

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  13.  $\begin{array}{r}7(2^{x-3})=56\end{array}$
  ::13. 7(2x-3)=56

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  14.  $\begin{array}{r}{16}^x\cdot 4^{x+1}={32}^{x+1}\end{array}$
  ::14. 16x4x+1=32x+1

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  15.  $\begin{array}{r}{3}^{3x+5}=3\cdot 9^{x+3}\end{array}$
  ::15. 33x+5=3x9x+3

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  Explore More
  ::探索更多

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  1. The value of a computer, $\begin{array}{r}C\end{array}$ , at the end of $\begin{array}{r}t\end{array}$  years is given by $\begin{array}{r}C=C_0(1-r{)}^t\end{array}$ , where $\begin{array}{r}{C}_0\end{array}$  is the original cost and $\begin{array}{r}r\end{array}$  is the rate of depreciation. Find the value of a computer at the end of 4 years if the original cost was $1,972, and the rate of depreciation is 10% . Round to the nearest cent.
  ::1. 计算机C在t年结束时的价值由C=C0(1-r)t给出,C0是原成本,r是折旧率。如果原成本为1,972美元,而折旧率为10%,则在4年结束时确定计算机的价值。

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  2. The function $\begin{array}{r}C(t)=C_0(1+r{)}^t\end{array}$  models the rise in the cost of a product that has a cost of $\begin{array}{r}{C}_0\end{array}$  today, subject to an average yearly inflation rate of $\begin{array}{r}r\end{array}$  for $\begin{array}{r}t\end{array}$  years. If the average annual rate of inflation over the next 11 years is assumed to be 3.5%, what will be the inflation-adjusted cost of a $27,000 car in 11 years? Round to the nearest cent.
  ::2. 函数C(t)=C0(1+r)t以今天成本为C0的产品的成本上涨为模型,但以t年平均年通货膨胀率r为单位。如果假设未来11年的平均年通货膨胀率为3.5%,那么11年中27 000美元的汽车按通货膨胀调整成本将是什么?回合到最近的百分比。

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  Answers for Review and Explore More Problems
  ::回顾和探讨更多问题的答复

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  Please see the Appendix. 
  ::请参看附录。

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  PLIX
  ::PLIX

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  Try this interactive that reinforces the concepts explored in this section:
  ::尝试这一互动,强化本节所探讨的概念: