Logarithmic & Exponential Functions (Cambridge (CIE) IGCSE Additional Maths): Exam Questions

For variables $x$ and $y$, plotting $\ln y$ against $\ln x$ gives a straight-line graph passing through the points $(6,5)$ and $(8,9)$.
Show that $y$ = ${\mathrm{e}}^p{x}^q$ where $p$ and $q$ are integers to be found.

Variables $x$ and $y$ are such that, when $\lg y$ is plotted against $x^3$, a straight line graph passing through the points (6,7) and (10,9) is obtained. Find $y$ as a function of $x$.

Given that ${\log }_{2}x+2{\log }_{4}y=8$, find the value of $xy$.

Variables $x$ and $y$ are connected by the relationship $y=A{x}^n$, where $A$ and $n$ are constants.

Transform the relationship $y=A{x}^n$ to straight line form.

When $\ln y$ is plotted against $\ln x$ a straight line graph passing through the points (0, 0.5) and (3.2, 1.7) is obtained.

Find the value of $n$ and of $A$.

Find the value of $y$ when $x=11$.

The population $P$, in millions, of a country is given by $P=A\times b^t$, where $t$ is the number of years after January 2000 and $A$ and $b$ are constants. In January 2010 the population was 40 million and had increased to 45 million by January 2013.

Show that $b=$1.04 to 2 decimal places and find $A$ to the nearest integer.

Find the population in January 2020, giving your answer to the nearest million.

In January of which year will the population be over 100 million for the first time?

The number, $b$, of bacteria in a sample is given by $b=P+Q{e}^{2t}$ , where $P$ and $Q$ are constants and $t$ is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week.

Find the value of $P$ and of $Q$.

Find the number of bacteria present after 2 weeks.

Find the first week in which the number of bacteria is greater than 1 000 000.

$\mathrm{f}\left(x\right)=3+{\mathrm{e}}^x\mathrm{for}x\in \mathrm{ℝ}$
$\mathrm{g}\left(x\right)=9x-5\mathrm{for}x\in \mathrm{ℝ}$

Find the exact solution of ${\mathrm{f}}^{-1}\left(x\right)=\mathrm{g}\text{'}\left(x\right)$ .

A mathematical model is used to predict the population of an island, $P$, over time, $t$ years.

A plot of $\ln P$ against $\ln t$ gives a straight line that is the perpendicular bisector of the points $\left(0,11\right)$ and $\left(6,7\right)$, as shown.

Find and simplify a formula for $P$ in terms of $t$.

Solve the equation $3^{2x+1}+8\left(3^x\right)–3=0$.

Solve the equation$4{\log }_{y}2+{\log }_{2}y=4$ .

Write the expression ${\log }_{a}9+\left({\log }_{a}b\right)\left({\log }_{\sqrt{b}}9a\right)$ in the form $c+d{\log }_{a}9$, where $c$ and $d$ are integers.

Find the exact solution of $3^{2x}-3^{x+1}-4=0$.

Solve the simultaneous equations

${10}^{x+2y}=5$,

${10}^{3x+4y}=50$ ,

giving $x$ and $y$ in exact simplified form.

${\log }_a\sqrt{b}-\frac{1}{2}={\log }_{b}a,\mathrm{where}a>0\mathrm{and}b>0$.

Solve this equation for $b$, giving your answers in terms of $a$.

Solve the simultaneous equations.

${\log }_3\left(x+y\right)=2$

$2{\log }_3\left(x+1\right)={\log }_3\left(y+2\right)$

It is known that $y=A\times {10}^{b{x}^2}$, where $A$ and $b$ are constants. When $\lg y$ is plotted against $x^2$, a straight line passing through the points (3.63, 5.25) and (4.83, 6.88) is obtained.

Find the value of $A$ and of $b$.

Using your values of $A$ and $b$, find the value of $y$ when $x=2$,

Find the positive value of $x$ when $y=4$.

${\log }_2\left(y+1\right)=3-2{\log }_{2}x$

${\log }_2\left(x+2\right)=2+{\log }_{2}y$

Show that $x^3+6x^2-32=0$.

Find the roots of $x^3+6x^2-32=0$.

Give a reason why only one root is a valid solution of the logarithmic equations. Find the value of $y$ corresponding to this root.