Indices, Surds and Logarithm
Question 1
(a)
Simplify $\frac{{{4}^{x+2}}-{{2}^{2x+1}}}{{{8}^{x}}\left( {{4}^{1-x}} \right)}$ and express it in the form $a\left( {{2}^{x-b}} \right)$ where $a$ and $b$ are integers.
[3]
(b)
Hence, solve for $x$ if $\frac{{{4}^{x+2}}-{{2}^{2x+1}}}{{{8}^{x}}\left( {{4}^{1-x}} \right)}=15$.
[2]
(a) 
(b) 
Question 2
(a)
Sketch the graph of $y=2-3\ln \left( x-1 \right)$, showing clearly the asymptote and giving the coordinates of the axes intercept(s) where applicable.
[4]
(b)
In order to solve the equation ${{e}^{\frac{x+1}{3}}}+1=x$, a graph of a suitable straight line is drawn on the same set of axes for $y=2-3\ln \left( x-1 \right)$. Find the equation of this line.
[3]
(a) 
(b) 
Question 3
The triangle $ABC$ is such that its area is $\frac{1}{4}\left( 12+9\sqrt{3} \right)$ cm$^{2}$, the length of $AB$ is $\left( \sqrt{3}+2 \right)$cm and angle $BAC$ is ${{60}^{\circ }}$. Without using a calculator, find
(i)
the length, in cm, of $AC$ in the form $a+b\sqrt{3}$, where $a$ and $b$ are integers,
(ii)
an expression, in cm$^{2}$, for $B{{C}^{2}}$ in the form $c+d\sqrt{3}$, where $c$ and $d$ are integers.
(i)
(ii) 
Question 4
Given ${{\log }_{12}}3=p$, express each of the following in terms of $p$.
(i)
${{\log }_{\sqrt{12}}}\frac{1}{9}$
(ii)
${{\log }_{3}}36$
(iii)
${{\log }_{144}}4$
(i)
(ii)
(iii) 
Question 5
Question 6
(i)
Sketch the graph of $y=\ln x$.
(ii)
In order to solve the equation ${{\text{e}}^{\text{e}x}}=x{{\text{e}}^{4}}$, a suitable straight line has to be drawn on the same set of axes as the graph of $y=\ln x$. Find the equation of the straight line and the number of solutions.
(i)
(ii)
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Question 7
Let $\text{f}\left( x \right)={{\log }_{a}}x$ and $\text{g}\left( x \right)={{\log }_{\frac{1}{a}}}x$, where base $a>1$, i.e. $0<\frac{1}{a}<1$.
(i)
Express $\text{g}\left( x \right)$ in terms of $\text{f}\left( x \right)$.
(ii)
Hence, sketch the graphs of $\text{f}\left( x \right)$ and $\text{g}\left( x \right)$ on the same diagram.
(i)
(ii)
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Question 8
2018 AM S4 MYE P2
On $6$ February 2016, a $6.4$-magnitude earthquake hit Pingtung city in Southern Taiwan. The magnitude, $M$, of an earthquake is measured on the Richter scale given by $M=\lg I-\lg S$ where $I$ is the intensity of the earthquake wave and $S$ is a constant value that represents the intensity of the smallest detectable wave as registered on a seismograph.
(i)
Express $\text{g}\left( x \right)$ in terms of $\text{f}\left( x \right)$.
In March $2011$, an earthquake hit Japan and the intensity of the earthquake wave was $1000$ times as strong as that of the $2016$ Taiwan earthquake.
(ii)
Find the magnitude of the $2011$ Japan earthquake on the Richter scale.
(i)
(ii)
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Question 9
2019 SNGS EOY Q5
Without using a calculator, find the value of ${{15}^{x}}$, given that ${{27}^{x}}\times {{25}^{2x}}={{3}^{2x+2}}\times {{5}^{3x-1}}$.
[4]
Question 10
2022 AHS AM S4 P1 Q4
Carbon-$14$ has a half-life of $5730$ years, which means that it takes $5730$ years for the original mass of Carbon-$14$ to be reduced by $50\%$. To determine the age of a plant or animal fossil, scientists determine the amount of Carbon-$14$ in the specimen as the Carbon-$14$ undergoes radioactive decay.
The amount of Carbon-$14$ in a piece of fossilised bone is given by $M={{M}_{0}}{{\text{e}}^{-kt}}$, where $k$ is a constant, $M$ is the mass of Carbon-$14$ in the specimen, ${{M}_{0}}$ is the initial mass of Carbon-$14$ and $t$ is measured in years.
(i)
Determine the value of $k$.
[2]
(ii)
Determine the percentage of Carbon-$14$ that has decayed in the specimen if the fossil is estimated to be about $900$ years old.
[2]
(iii)
Express the rate of change of percentage of Carbon-$14$ in terms of $t$.
[2]
(i) 
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(ii) 
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(iii) 
Question 11
2022 FMSS AM S4 P2 Q7
(a)
Solve the simultaneous equations
$\text{e}\sqrt{{{\text{e}}^{x}}}={{\text{e}}^{2y}}$,
${{\log }_{4}}\left( x+2 \right)=1+{{\log }_{2}}y$.
[8]
(b)
Solve the equation $2\left( {{100}^{y}} \right)-{{10}^{y}}=6$.
[4]
(a)
(b)
Question 12
Find the possible values of the real numbers of $a$ and $b$ such that $\left( a-6\sqrt{5} \right)\left( 2+b\sqrt{5} \right)=-82$.
Question 13
Given that ${{\log }_{3}}x=a$, ${{\log }_{9}}y=b$ and $x{{y}^{2}}=81$, $\frac{{{x}^{2}}}{y}=\frac{1}{3}$, find the value of $a$ and of $b$.
Question 14
Given that ${{\log }_{2}}x=p$ and ${{\log }_{4}}y=q$, express ${{x}^{2}}y$ and $\frac{x}{{{y}^{2}}}$ as powers of $2$. Further given that ${{x}^{2}}y=64$ and $\frac{x}{{{y}^{2}}}=2$, determine the value of $p$ and of $q$.
Question 15
Knowing the rate at which fire grows helps the Singapore Civil Defence Force determine the size of fire and plan for safe evacuation. The area damaged, $A$m$^{2}$ in $t$ minutes since the start of the fire is given by the formula $A={{A}_{0}}{{e}^{0.5t}}$, where ${{A}_{0}}$m$^{2}$ is the initial area that caught fire.
(a)
An initial area of $36.5$ m$^{2}$ caught fire in a warehouse. The fire continued to burn for $10$ minutes. Calculate the area damaged by the fire at the end of the duration.
[1]
(b)
An area of $2$m$^{2}$ caught fire and by the time the fire was put out, $36$km$^{2}$ of land had been damaged by fire. Calculate the duration, in minutes, the fire was burning before it was finally put out. Leave your answers to the nearest whole number.
[$1$km$^{2}$$=1\text{ }000\,000$m$^{2}$]
[3]
(a)
(b)
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Question 16
Solve the following simultaneous equations
$9\left( {{3}^{x}} \right)={{81}^{y}}$,
${{\log }_{2}}\left( x+2 \right)=3+6{{\log }_{8}}y$.
