Course: 8.10 解决指数等号

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解决指数等号
"I'm thinking of a number," you tell your best friend. "The number I'm thinking of satisfies the equation $\begin{align*}4^{x + 1} = 256\end{align*}$ . What number are you thinking of?
::"我在想一个数字",你告诉你最好的朋友。"""我所想的满足4x+1=256方程式的数字是多少?

Exponential Equations
::指数等量

Until now, we have only solved pretty basic exponential equations. But, what happens when the power is not easily found? We must use logarithms, followed by the Power Property to solve for the exponent .
::到目前为止,我们只解决了相当基本的指数方程式。但是,当电力不易找到时会怎样呢?我们必须使用对数,然后是电力财产,为指数者解决。

Let's solve the following more complicated exponential equations.
::让我们解决以下更复杂的指数方程式。

$\begin{align*}6^x=49\end{align*}$ (Round your answer to the nearest three decimal places)
::6x=49( 将您的答复锁定到最接近小数点后三位位数)

To solve this exponential equation, let’s take the logarithm of both sides. The easiest logs to use are either $\begin{align*}\ln\end{align*}$ (the natural log), or log (log, base 10). We will use the natural log.
::为了解决这个指数式方程式,让我们使用两边的对数。最容易使用的日志是 in( 自然日志 ) , 或者 log( 基数 10 ) 。我们将使用自然日志。

$\begin{aligned} 6^{x} & = 49 \\ \ln 6^{x} & = \ln 49 \\ x \ln 6 & = \ln 49 \\ x & = \frac{\ln 49}{\ln 6} \approx 2.172 \end{aligned}$
$\begin{align*}6^x &= 49 \\ \ln 6^x &= \ln 49 \\ x \ln 6 &= \ln 49 \\ x &= \frac{\ln 49}{\ln 6} \approx 2.172\end{align*}$
::6x=49ln_6x=ln_49xln_9x_6=ln_49x=ln_49_6__6}#2.172

$\begin{align*}10^{x-3}=100^{3x+11}\end{align*}$
::10x-3=1003x+11

Change 100 into a power of 10.
::将100变成10的功率

$\begin{aligned} 10^{x - 3} & = 10^{2 (3 x + 11)} \\ x - 3 & = 6 x + 22 \\ - 25 & = 5 x \\ - 5 & = x \end{aligned}$
$\begin{align*}10^{x-3} &= 10^{2(3x+11)} \\ x-3 &= 6x+22 \\ -25 &= 5x \\ -5 &= x\end{align*}$
::10x-3=102(3x+11)x-3=6x+22-2-5=5x-5=x

$\begin{align*}8^{2x-3}-4=5\end{align*}$
::82x-3-4=5

Add 4 to both sides and then take the log of both sides.
::双方加4,然后取出双方的日志。

$\begin{aligned} 8^{2 x - 3} - 4 & = 5 \\ 8^{2 x - 3} & = 9 \\ \log 8^{2 x - 3} & = \log 9 \\ (2 x - 3) \log 8 & = \log 9 \\ 2 x - 3 & = \frac{\log 9}{\log 8} \\ 2 x & = 3 + \frac{\log 9}{\log 8} \\ x & = \frac{3}{2} + \frac{\log 9}{2 \log 8} \approx 2.56 \end{aligned}$
$\begin{align*}8^{2x-3}-4 &= 5 \\ 8^{2x-3} &= 9 \\ \log 8^{2x-3} &= \log 9 \\ (2x-3)\log 8 &= \log 9 \\ 2x-3 &= \frac{\log 9}{\log 8} \\ 2x &= 3 + \frac{\log 9}{\log 8} \\ x &= \frac{3}{2}+\frac{\log 9}{2 \log 8} \approx 2.56\end{align*}$
::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. 56

Notice that we did not find the numeric value of $\begin{align*}\log9\end{align*}$ or $\begin{align*}\log8\end{align*}$ until the very end. This will ensure that we have the most accurate answer.
::注意我们直到最后才发现对数=%9 或对数=8。这将确保我们得到最准确的答案。

Examples
::实例

Example 1
::例1

Earlier, you were asked to determine what number you are thinking of if the number satisfies the equation $\begin{align*}4^{x+1} = 256\end{align*}$ .
::早些时候,有人要求你确定,如果数字满足4x+1=256等式,你考虑的数字是多少。

We can rewrite the equation $\begin{align*}4^{x + 1} = 256\end{align*}$ as $\begin{align*}2^{2(x+1)} = 2^8\end{align*}$ and solve for x .
::我们可以重写 4x+1=256 的方程式为 22(x+1)=28, 并解决 x 。

$\begin{aligned} 2^{2 (x + 1)} = 2^{8} \\ 2^{2 x + 2} = 2^{8} \\ 2 x + 2 = 8 \\ x = 3 \end{aligned}$
$\begin{align*}2^{2(x+1)} = 2^8\\ 2^{2x +2} = 2^8\\ 2x + 2 = 8\\ x = 3\end{align*}$
::22(x+1)=2822x+2=282x+2=8x=3

Therefore , you're thinking of the number 3.
::因此,你是在考虑3号

Example 2
::例2

Solve: $\begin{align*}4^{x-8}=16\end{align*}$ .
::解决:4x-8=16。

Change 16 to $\begin{align*}4^2\end{align*}$ and set the exponents equal to each other.
::更改 16 到 42 , 并将指数等值设定。

$\begin{aligned} 4^{x - 8} & = 16 \\ 4^{x - 8} & = 4^{2} \\ x - 8 & = 2 \\ x & = 10 \end{aligned}$
$\begin{align*}4^{x-8} &= 16 \\ 4^{x-8} &= 4^2 \\ x-8 &= 2 \\ x &=10\end{align*}$
::4x-8=164x-8=42x-8=2x=10

Example 3
::例3

Solve: $\begin{align*}2(7)^{3x+1} =48\end{align*}$ .
::解决时间: 2(7)3x+1=48。

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

$\begin{aligned} 2 (7)^{3 x + 1} & = 48 \\ 7^{3 x + 1} & = 24 \\ \ln 7^{3 x + 1} & = \ln 24 \\ (3 x + 1) \ln 7 & = \ln 24 \\ 3 x + 1 & = \frac{\ln 24}{\ln 7} \\ 3 x & = - 1 + \frac{\ln 24}{\ln 7} \\ x & = - \frac{1}{3} + \frac{\ln 24}{3 \ln 7} \approx 0.211 \end{aligned}$
$\begin{align*}2(7)^{3x+1} &= 48 \\ 7^{3x+1} &= 24 \\ \ln 7^{3x+1} &= \ln 24 \\ (3x+1)\ln 7 &= \ln 24 \\ 3x+1 &= \frac{\ln 24}{\ln 7} \\ 3x &= -1 + \frac{\ln 24}{\ln 7} \\ x &= -\frac{1}{3} + \frac{\ln 24}{3 \ln 7} \approx 0.211\end{align*}$
::2(7)3x+1=4873x+1=24ln 73x+1=ln 24(3x+1) 7=ln 2443x+1=ln 24l_73x_1+1+ln 2400=7x=13+ln 243ln @7_0

Example 4
::例4

Solve: $\begin{align*}\frac{2}{3} \cdot 5^{x+2}+9=21\end{align*}$ .
::解决时间:235x+2+9=21。

Subtract 9 from both sides and multiply both sides by $\begin{align*}\frac{3}{2}\end{align*}$ . Then, take the log of both sides.
::从两边减9, 乘以两边32, 然后,取取两边的日志。

$\begin{aligned} \frac{2}{3} \cdot 5^{x + 2} + 9 & = 21 \\ \frac{2}{3} \cdot 5^{x + 2} & = 12 \\ 5^{x + 2} & = 18 \\ (x + 2) \log 5 & = \log 18 \\ x & = \frac{\log 18}{\log 5} - 2 \approx - 0.204 \end{aligned}$
$\begin{align*}\frac{2}{3} \cdot 5^{x+2}+9 &= 21 \\ \frac{2}{3} \cdot 5^{x+2} &= 12 \\ 5^{x+2} &= 18 \\ (x+2)\log 5 &= \log 18 \\ x &= \frac{\log 18}{\log 5}-2 \approx -0.204\end{align*}$
::23_5x+2+9=2123_5x+2=125x+2=18(x+2),log_5=log_18x=log_18log_5-20.204

Review
::回顾

Use logarithms and a calculator to solve the following equations for $\begin{align*}x\end{align*}$ . Round answers to three decimal places.
::使用对数和计算器解析 x. x 的下列方程式。圆形答案为小数点后三个位数。

$\begin{align*}5^x = 65\end{align*}$
::5x=65

$\begin{align*}7^x = 75\end{align*}$
::7x=75

$\begin{align*}2^x = 90\end{align*}$
::2x=90

$\begin{align*}3^{x-2} = 43\end{align*}$
::3x-2=43

$\begin{align*}6^{x+1}+3=13\end{align*}$
::6x+1+3=13

$\begin{align*}6(11^{3x-2})=216\end{align*}$
::6(113x-2)=216

$\begin{align*}8+13^{2x-5}=35\end{align*}$
::8+132x-5=35

$\begin{align*}\frac{1}{2} \cdot 7^{x-3}-5=14\end{align*}$
::12-7x-3-5=14

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。

$\begin{align*}4^x=8\end{align*}$
::4x=8

$\begin{align*}9^{x-2} = 27\end{align*}$
::9x-2=27

$\begin{align*}5^{2x+1}=125\end{align*}$
::52x+1=125

$\begin{align*}9^3=3^{4x-6}\end{align*}$
::93=34x-6

$\begin{align*}7(2^{x-3})=56\end{align*}$
::7(2x-3)=56

$\begin{align*}16^x \cdot 4^{x+1}=32^{x+1}\end{align*}$
::16x4x+1=32x+1

$\begin{align*}3^{3x+5}=3 \cdot 9^{x+3}\end{align*}$
::33x+5=3x9x+3

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

解决指数等号

Exponential Equations ::指数等量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Exponential Equations
::指数等量

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)