Course: 16.4 答复----第4章:指数和对数函数

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回答 - 第4章:指数和对数函数
Section 4.2: Graphing and Evaluating Exponential Functions
::第4.2节:绘制图表和评估指数职能

$\begin{aligned} y = 5 \cdot 5^{x} \end{aligned}$
::y=5=5=5x

$\begin{aligned} y = 2 \cdot 4^{x} \end{aligned}$
::y=2×4x y=2×4x

$\begin{aligned} y = 16 \cdot 3^{x} \end{aligned}$
::y=16=3x

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

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

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

$\begin{aligned} f (x) = 3^{x - 4} \end{aligned}$ will be moved to the right 4 spaces.
::f(x) = 3x-4 将移到右边的 4 空格。

::f(x) = 3x-4 将移到右边的 4 空格。

$\begin{aligned} f (x) = - 4^{x} \end{aligned}$ will be reflected across the $\begin{aligned} x \end{aligned}$ axis.
::f(x) =-4x 将反射到 x 轴。

::f(x) =-4x 将反射到 x 轴。

$\begin{aligned} f (x) = 3^{x} - 2 \end{aligned}$ will be moved down 2 spaces.
::f(x)=3x-2 将会向下移动两个空格。

::f(x)=3x-2 将会向下移动两个空格。

$\begin{aligned} f (x) = {- 5}^{x + 2} \end{aligned}$ will move left 2 spaces, and be reflected over the $\begin{aligned} x \end{aligned}$ axis.
::f(x) =-5x+2 将左移两个空格,并反映在 x 轴上方。

::f(x) =-5x+2 将左移两个空格,并反映在 x 轴上方。

$\begin{aligned} f (x) = 5^{x - 4} - 3 \end{aligned}$ will be moved down 3 spaces and right 4 spaces.
::f(x) = 5x-4-3 将会向下移动 3 个空格和右 4 个空格。

::f(x) = 5x-4-3 将会向下移动 3 个空格和右 4 个空格。

Because the $\begin{aligned} y \end{aligned}$ -intercept occurs where $\begin{aligned} x = 0 \end{aligned}$ and $\begin{aligned} y = a \cdot b^{0} \end{aligned}$ → $\begin{aligned} y = a \cdot 1 \end{aligned}$ → $\begin{aligned} y = a \end{aligned}$ .
::因为y- interview 发生于 x=0 和 y=a&b0 y=a1 y=a。

::因为y- interview 发生于 x=0 和 y=a&b0 y=a1 y=a。

10,485,760 rabbits
::10 485 760只兔子

Approximately 1.138 grams
::约1 138克

Section 4.3: Graphing and Evaluating Logarithmic Functions
::第4.3节:绘图和评估对数函数

You could graph it as $\begin{aligned} \frac{log (x)}{log (4)} \end{aligned}$ .

::您可以用对数(x)log(4)来图示它。

Yes
::是是

No
::否无

No
::否无

4.4 Properties of Logs
::4.4 日志属性

False
::假假

False
::假假

True
::真实

$\begin{aligned} \log (8 x^{2} + 16 x) \end{aligned}$
::log ( 8x2+16x)

$\begin{aligned} \log x^{6} \end{aligned}$
::对数============================================================================================

$\begin{aligned} \log_{2} (\frac{3 x^{4}}{y}) \end{aligned}$
::log2( 3x4y)

$\begin{aligned} \log_{3} z^{4} \end{aligned}$
::对数 3z4

$\begin{aligned} \log_{4} 2 + 3 \log_{4} x - \log_{4} 5 \end{aligned}$
::对数 4 @% 2+3log4_x-log4_% 5

$\begin{aligned} \ln 4 + \ln x + 2 \ln y - \ln 15 \end{aligned}$
::In4+lnx+2lny-ln15

$\begin{aligned} 2 \log x + 3 \log y + 3 \log z - \log 3 \end{aligned}$
::2log_x+3log_y+3log_z_log%3

$\begin{aligned} \log_{2} 10 = x + 1 \end{aligned}$
::对数 2\\\\\\ 10=x+1

$\begin{aligned} 2^{12} = x - 1 \end{aligned}$
::212=x-1

Let $\begin{aligned} y = \log_{b^{n}} x \end{aligned}$ . Then $\begin{aligned} {(b^{n})}^{y} = x \end{aligned}$ and $\begin{aligned} b^{n y} = x \end{aligned}$ . Take the log base b of both sides, and you have $\begin{aligned} (n y) \log_{b} b = \log_{b} x \end{aligned}$ . Simplify to get $\begin{aligned} n y = \log_{b} x \end{aligned}$ . Solve for $\begin{aligned} y, \end{aligned}$ and you have $\begin{aligned} y = \frac{1}{n} \log_{b} x \end{aligned}$ . Since both $\begin{aligned} \frac{1}{n} \log_{b} x \end{aligned}$ and $\begin{aligned} \log_{b^{n}} x \end{aligned}$ are equal to $\begin{aligned} y \end{aligned}$ , they are equal to each other.
::Let y=logbnx. 然后 (bn)y=x 和 bny=x. 使用两边的对数基准 b, 您有( ny) logbb=logbx. 简化以获得 ny=logbx. solve y, 您有 y=1nlogbx. 由于 1nlogbx 和 logbnx 都等于 y, 它们彼此相等。

From the previous problem, you know that $\begin{aligned} \log_{b^{n}} (x^{n}) = \frac{1}{n} \log_{b} x^{n} . \end{aligned}$ Use the exponentiation property, and you have $\begin{aligned} \frac{1}{n} \log_{b} x^{n} = n (\frac{1}{n}) \log_{b} x, \end{aligned}$ which simplifies to $\begin{aligned} \log_{b} x \end{aligned}$ .
::从上一个问题中, 您知道对数 {( xn) =1nlogb}}xn。使用引号属性, 您有 1nlogb}xn =n(1n) logb}x, 它简化为对数 {x 。

$\begin{aligned} y = \log_{\frac{1}{b}} \frac{1}{x} . \end{aligned}$ Let $\begin{aligned} y = \log_{\frac{1}{b}} \frac{1}{x} \end{aligned}$ . Then $\begin{aligned} {(\frac{1}{b})}^{y} = \frac{1}{x} \end{aligned}$ . You can rewrite this equation as $\begin{aligned} x = b^{y} . \end{aligned}$ You can then rewrite in logarithmic form as $\begin{aligned} \log_{b} x = y \end{aligned}$ . Since both $\begin{aligned} \log_{b} x \end{aligned}$ and $\begin{aligned} \log_{\frac{1}{b}} \frac{1}{x} \end{aligned}$ are equal to $\begin{aligned} y \end{aligned}$ , they are equal to each other.
::y=log1b1x. 让我们y=log1b1x. 然后 (1b)y=1x. 您可以重写此方程式为 x=by。然后您可以重写对数格式为logbx=y。由于logbx和log1b1x都等于y, 它们彼此等同。

Section 4.5: Solving Exponential Equations Using Logs
::第4.5节:使用日志解决指数等同

$\begin{aligned} x = 1.292 \end{aligned}$
::x=1.292x=1.292

$\begin{aligned} x = 0.431 \end{aligned}$
::x=0.431xx=0.431

$\begin{aligned} x = 0.697 \end{aligned}$
::x=0.697xx=0.697

$\begin{aligned} x = 1.333 \end{aligned}$
::x=1.333x=1.333

$\begin{aligned} x = 3.754 \end{aligned}$
::x=3.754x=3.754

$\begin{aligned} x = 0.783 \end{aligned}$
::x=0.783x=0.783

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

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

$\begin{aligned} x = 3.0259 \end{aligned}$
::x=3.0259

$\begin{aligned} x = 8.9872 \end{aligned}$
::x8.9872x8.9872

$\begin{aligned} x = \frac{\log (c + a)}{\log b} \end{aligned}$
::x=log(c+a)logb

$\begin{aligned} x = - 0.057 \end{aligned}$
::x=- 0.057

$\begin{aligned} x = 1.8 \end{aligned}$
::x=1.8x=1.8

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

$\begin{aligned} x = 27 \end{aligned}$
::x=27x=27

Section 4.6: Solving Logarithmic Equations
::第4.6节:解决对数等数

$\begin{aligned} x = 32, 768 \end{aligned}$
::x=32,768x=32,768

$\begin{aligned} x \approx 498.831 \end{aligned}$
::x498.831

$\begin{aligned} x = 86 \end{aligned}$
::x=86x=86

$\begin{aligned} x = 170 \end{aligned}$
::x=170 x=170

$\begin{aligned} x \approx 1.132 \end{aligned}$
::x1132

$\begin{aligned} x = \sqrt[3]{3} \approx 1.442 \end{aligned}$
::x=33=1.442

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

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

$\begin{aligned} x = 8 \end{aligned}$
::x=8x=8

$\begin{aligned} x = 27 \end{aligned}$
::x=27x=27

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

$\begin{aligned} x \approx 3.272 \end{aligned}$
::x*3.272

$\begin{aligned} x = \frac{4}{3} \end{aligned}$
::x=43 x=43

$\begin{aligned} x = 2 + 2 \sqrt{11} \approx 8.633 \end{aligned}$
::x=2+211=8.633

$\begin{aligned} x = 6 \end{aligned}$
::x=6x=6

Section 4.7: Compound Interest
::第4.7节:复合利息

The compound interest formula is $\begin{aligned} A (t) = p {(1 + r)}^{t} . \end{aligned}$
::复合利息公式为 A(t)=p(1+r)t。

They would have earned $373.82.
::他们本可以挣373.82美元。

Kyle's balance would be $1,098.81.
::Kyle的余额将是1 098.81美元。

Yearly simple interest is effectively the same as compound interest compounding yearly. Roberta would have $18,022.10 in the account.
::一年简单利息实际上与每年复利复利相同,罗伯塔账户中有18 022.10美元。

The bank has been paying approximately 3.5% annually.
::银行每年支付约3.5%。

She could expect to have $3,122.
::她可以指望有3 122美元。

There is a balance of $885.08.
::余额为885.08美元。

Her original deposit was $650.
::她原先的存款是650美元

$\begin{aligned} 1000 ({1 + .05)}^{7} \end{aligned}$

$\begin{aligned} f (x) = 3000 {(1.14}^{x}) \end{aligned}$
$7,500
:xx) = 30000(1.14x) 7 500美元

He will owe a total of $318.27.
::他将共欠318.27美元。

$600

$715.56

$15,300.14

$13,138.75

Section 4.8: Population Growth Models and Logistic Functions
::第4.8节:人口增长模式和后勤职能

$\begin{aligned} f (x) = \frac{12}{1 + 1.4 \cdot {0.51}^{x}} \end{aligned}$
::f(x) = 121+1.40. 51x

$\begin{aligned} f (x) = \frac{200}{1 + \frac{1}{3} \cdot {0.8027}^{x}} \end{aligned}$
::f(x) = 2001+130. 8027x

$\begin{aligned} f (x) = \frac{1, 500}{1 + 9 \cdot {0.749}^{x}} \end{aligned}$
::f(x)=1,51+90.749x

$\begin{aligned} f (x) = \frac{1, 000, 000}{1 + 9 \cdot {0.959}^{x}} \end{aligned}$
::f(x) = 1 000 0001+90. 959x

$\begin{aligned} f (x) = \frac{30, 000, 000}{1 + 2.75 \cdot {0.89}^{x}} \end{aligned}$
::f(x)=30 000 0001+2.750.89x

32

8

25

5

4

$\begin{aligned} \frac{4}{3} \end{aligned}$

Any function in the form $\begin{aligned} f (x) = \frac{c}{1 + a \cdot b^{- x}}, \end{aligned}$ where $\begin{aligned} 0 < b < 1 \end{aligned}$ .
::窗体 f(x) = c1+ab-x 中的任何函数,其中 0<b> 1。

Section outline

General

回答 - 第4章:指数和对数函数

Section 4.2: Graphing and Evaluating Exponential Functions ::第4.2节:绘制图表和评估指数职能

Section 4.3: Graphing and Evaluating Logarithmic Functions ::第4.3节:绘图和评估对数函数

4.4 Properties of Logs ::4.4 日志属性

Section 4.5: Solving Exponential Equations Using Logs ::第4.5节:使用日志解决指数等同

Section 4.6: Solving Logarithmic Equations ::第4.6节:解决对数等数

Section 4.7: Compound Interest ::第4.7节:复合利息

Section 4.8: Population Growth Models and Logistic Functions ::第4.8节:人口增长模式和后勤职能

Section 4.2: Graphing and Evaluating Exponential Functions
::第4.2节:绘制图表和评估指数职能

Section 4.3: Graphing and Evaluating Logarithmic Functions
::第4.3节:绘图和评估对数函数

4.4 Properties of Logs
::4.4 日志属性

Section 4.5: Solving Exponential Equations Using Logs
::第4.5节:使用日志解决指数等同

Section 4.6: Solving Logarithmic Equations
::第4.6节:解决对数等数

Section 4.7: Compound Interest
::第4.7节:复合利息

Section 4.8: Population Growth Models and Logistic Functions
::第4.8节:人口增长模式和后勤职能