Math topic中文

Functions and graphs

exponential and logarithmic functions $a^{x}$ , $e^{x}$ , $\log_{a}x$ , $\ln x$ and their graphs, including:

laws of logarithms
equivalence of $y = a^{x}$ and $x = \log_{a}y$
change of base of logarithms

Practice questions

1Mr Phua bought a new car. After $t$ months, its value $V$ is given by $V = 250000e^{-pt}$ , where $p$ is a constant. The…
2Find the gradient function of the curve $y = 2x^2 - \ln x + c$ . The curve touches the positive $x$ -axis at point $T$ . Fi…
3The point $P$ lies on the curve $y = \ln\left(\frac{x+1}{x-1}\right)$ for $x > 1$ . The normal to the curve at $P$ is par…
4The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
5函数 $f\left(x\right)=\begin{cases}{2}^{x}-2\;,\;x\leqslant 1\;,\;\\ {\log}_{2}\left(x-1\right)\;,\;x>1\end{cases}\;$ 则 $f\left[f\left(\dfrac{5}{2}\right)\right]=$ …
6设函数 $f\left(x\right)$ 是定义在 $R$ 上的奇函数，且对任意 $x∈R$ 都有 $f\left(x\right)=f\left(x+4\right)$ ，当 $x∈\left(0,2\right)$ 时， $f\left(x\right)={2}^{x}$ …
7Given that $\sqrt{125^x} = \dfrac{5^{1-x}}{25}$ , find the value of $\sqrt{125^x}$ . Show that $\log_2 x - \log_{16} x = \dfrac{3\lg x}{4\lg 2}$ …
8The solution of the equation $3^{2x-1}=4(3^x)$ can be expressed as $\dfrac{\ln p}{\ln q}$ . Find the value of $p$ and o…
9若“ $m>a$ ”是“函数 $f(x)=\:(\dfrac{1}{3})^{x}+m-\dfrac{1}{3}$ 的图象不过第三象限”的必要不充分条件，则实数 $a$ 能取的最大整数为___.
10Express $\frac{8x+13}{(1+2x)(2+x)^2}$ in partial fractions. Express $\frac{8x+13}{(1+2x)(2+x)^2}$ in partial fractions…
11Given that $u = 2^x$ , express $2^{x+1} = 5(2^x) + 3$ as an equation in $u$ . Hence find the value of $x$ for which $2^{x+1} = 5(2^x) + 3$ …
12The equation of a curve is given by $y = 5xe^{2x+1}$ . Find $\frac{dy}{dx}$ . Hence, find $\int xe^{2x+1} \text{ d}x$ . Det…
13Use the substitution $u = 7^x$ , solve the equation $7^{2x+2}-4(7^{x+1})-5=0$ and show that the solution may be written i…
14化简求值： $(1)0.{064}^{- \frac{1}{3}}-( \dfrac{1}{8}{)}^{0}+{16}^{ \frac{3}{4}}+0.{25}^{ \frac{1}{2}} (2)\dfrac{1}{2} \lg 25+\lg 2+( \dfrac{1}{3}{)}^{{\log }_{3}2} -\log _{2}9×\log _{3}2.$ …
15Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
16Differentiate $\ln\left(\frac{3x}{x^2+1}\right)$ with respect to $x$ . Hence find $\displaystyle \int \frac{2x}{x^2+1}\,dx$ …
17Given that $u = \left(\dfrac{3}{2}\right)^x$ , express $9^x + 6^x = 6(4^x)$ as a quadratic equation in $u$ . Hence, or oth…
18Solve the equation $x + 1 + \sqrt{5 - x} = 0$ . Solve the equation $\log_x(2x-1) = \log_{x^2}(4-5x)$ .
19Express $\frac{2x+15}{(3+x)(3-2x)}$ in partial fractions. Hence evaluate $\int_{0}^{1} \frac{2x+15}{(3+x)(3-2x)} \, dx$ .
20已知 $m>1$ ， $n>1$ ，且 $\ln m-\ln n=2n-m$ ，下列结论正确的是 $(\quad)$ ① $(\dfrac{1}{2})^{m}< (\dfrac{1}{2})^{n}$ ；② $\dfrac{n}{m}>\dfrac{n+1}{m+1}$ …
21The equation of a curve is $y = e^{-2x}\tan x$ . Show that $\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2$ . Find, in terms of $\pi$ …
22The point A lies on the curve $y = x \ln x^2$ , $x > 0$ . The tangent to the curve at A is parallel to the line $y - 3x = 1$ …
23Solve the equation $\log_3 x - 3\log_x 9 = 1$ .
24Solve the equation $2^{4p+1} + 20(4^{p-1}) = 3$ . Explain why $2^{4p+1} + 20(4^{p-1}) = k$ has no solution if $k < -3\frac{1}{8}$ …
25若函数 $f(x)=\dfrac{e^{x-1}}{x}$ ， $g(x)=\dfrac{x}{1+\ln x}$ ，存在 $x_{1}$ 、 $x_{2}$ 使得 $f({x}_{1})=g({x}_{2})$ ，则下列说法不正确的是 $(\quad)$ …
26By using an appropriate substitution, solve the equation $e^{\frac{1}{2}x} = 2 + 24 e^{-\frac{1}{2}x}$ .
27Show that the equation $3e^{2x}+7e^{x}=6$ has only one solution and find its value correct to 2 significant figures. Fac…
28Measured values of $x$ and $y$ are given in the following table. | $x$ | 1.2 | 1.6 | 2.0 | 2.6 | 3.5 | 4.5 | |---|---|--…
29Find the value of $10^x$ , given that $4^{x+1} \times 5^{2x-3} = 10^{3x}$ . Hence, solve $4^{x+1} \times 5^{2x-3} = 10^{3x}$ …
30The table shows experimental values of $x$ and $y$ . | $x$ | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---| | $y$ | 11.8 |…
31在 $y=3^{x}$ ， $y=\log _{0.3}x$ ， $y=x^{3}$ ， $y= \sqrt {x}$ ，这四个函数中当 $0 < x_{1} < x_{2} < 1$ 时，使 $f( \dfrac {x_{1}+x_{2}}{2}) < \dfrac {f(x_{1})+f(x_{2})}{2}$ …
32A curve is defined by the equation $y = \ln\left(\frac{\text{e}^x}{x^2 + 10}\right)$ . Show that the curve has no turning…
33Solve the following equations. $5^x - 5^{\frac{x}{2}+1} = 6$ , $2\lg(x - 3) - \lg(x + 7) = \frac{1}{\log_{100} 10}$ .
34The population of bacteria, $P$ , after $t$ hours can be modelled by $P = P_0(2^t)$ , where $P_0$ is a constant. Find the…
35Sketch the graph of $y = 3^x$ . In order to solve the equation $\log_9(5-x) = \frac{x}{2}$ , a suitable straight line has…
36A metal cube is heated to a temperature of $212^\circ\text{C}$ before being dropped into a liquid. As the cube cools, it…
37已知函数 $f(x)=2x^{2}-4x+a$ ，函数 $g(x)=\log_{a}x(a>0,$ 且 $a≠1).(1)$ 若函数 $f(x)$ 在区间 $[-1,2m]$ 上不具有单调性，求实数 $m$ 的取值范围； (2)若 $f(1)=g(1).\:$ …
38Show that the solution of the equation $2^{3x+4}\times 5^{2x-1}=16^{x}\times 5^{3x}$ is $\lg\dfrac{16}{5}$ . Express $2\log_{2}x-\log_{2}(x-4)=3$ …
39A pot of melted chocolate is cooled from its initial temperature to a temperature of $T^\circ\text{C}$ in $x$ minutes an…
40In an experiment, 5 million cells were stored in a container. It was observed that 40% of the cells are dying every minu…
41Find the derivative of the functions $y = \ln x$ and $y = \ln 2x$ respectively. Explain why, in (i), having the same der…
42The table below shows experimental values of two variables $x$ and $y$ . It is known that $x$ and $y$ are related by an e…
43Solve the equations. $\log_2(x+4) = 2\log_2 x - 1$ $10\log_y 5 + 3 = \log_5 y$
44若曲线 $y=e^{-x}$ 上点 $P$ 处的切线平行于直线 $2x+y+1=0$ ，则点 $P$ 的坐标为 ___ ．
45Sketch the graph of of $y = \log_2(3x + 1)$ and label the point where $x = \frac{1}{3}$ . Explain why $x > -\frac{1}{3}$ .…
46Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
47The diagram shows part of the curve $y = \frac{1}{4}x^2 - \frac{1}{2}\ln\left(\frac{x}{3}\right)$ and $P(3,\,2.25)$ is…
48The number, n, of a certain type of bacteria increases with time t minutes. Recorded values of n and t are shown in the…
49 $a\otimes b=\begin{cases} b, \\ a, \\ \end{cases}\begin{matrix} a\geqslant b \\ a < b \\ \end{matrix}f\left( x \right)={{3}^{-x}}\otimes {{3}^{x}}$ …
50Given that $\log_a 8 + 3\log_a N = 3$ and $N > \frac{1}{4}$ , find the range of values of $a$ . Solve the equation $\log_3 x - \log_x 3x - 2 = 0$ …
51Express $\dfrac{8 - 3x}{(2x + 3)(x^2 + 4)}$ in partial fractions. Differentiate $\ln\left(x^2 + 4\right)$ with respect t…
52The approximate mean distance $x$ (in millions of kilometres) from the centre of the Sun and the period of the orbit $T$ …
53Given that $\frac{(\log_y x)^2}{\log_x y} + 64 = 0$ , express $y$ in terms of $x$ . Solve the equation $\log_2 x + 3 = 2\log_2(x - 1)$ …
54Solve $9^{x+1} - 3^{x+2} = 2\left(3^{x+1} - 2\right)$ . Given that $\log_3(2x-1) = \log_9(2x-1) + 1$ , find the value of $x$ …
55函数 $f(x)={3}^{-{x}^{2}+2ax}$ 在区间 $(-∞,1)$ 内递增，则 $a$ 的取值范围是___。
56关于函数 $f(x)=\begin{cases}{2^{x}-a,0\leqslant x< 2}\\ {b-x,x\geqslant 2}\end{cases}$ ，其中 $a$ ， $b\in R$ ，给出下列四个结论：甲： $6$ …
57Show that the equation $2(e^x - 3) = e^{\frac{1}{2}x}$ has only one solution and find this value correct to 3 significan…
58Given that $\log_x p = 9$ and $\log_x q = 6$ , find the value of $\log_{\frac{p}{q}} x^2$ . Solve the equation $\lg(3x + 1) + \lg(2x - 1) = 2 - \lg 2$ …
59设 $a={2020}^{ \frac{1}{2019}},b=lo{g}_{2019} \sqrt{2020},c=lo{g}_{2020} \sqrt{2019}$ ，则 $(\:\:\:\:)$ A. $c > b > a$ B…
60The variables $x$ and $y$ are defined such that $\log_{2} x - 3\log_{4} y = \log_{9} 3$ . Give a reason why $x$ and $y$ m…