Logarithmic and exponential functions

Assume that $k$ is a positive constant.

Feb/March 2016

Solve for $x$ in the equation $2\ln(2x) - \ln(x + 3) = \ln(3x + 5)$.

Feb/March 2017

Solve for $x$ in the inequality $|2x - 5| < |x + 3|$.

Feb/March 2017

The polynomial $p(x)$ is given by $p(x) = 4x^3 + 4x^2 - 29x - 15$.

Feb/March 2018

The variables $x$ and $y$ are related by $y = A e^{px+p}$, with $A$ and $p$ as constants. A plot of $\ln y$ against $x$ forms a straight line through $(1, 2.835)$ and $(6, 6.585)$, as illustrated.

Feb/March 2019

The polynomial $p(x)$ is given by $p(x)=x^3+ax+b$, with $a$ and $b$ as constants. You are told that $(x+2)$ is a factor of $p(x)$ and that, on dividing $p(x)$ by $(x-3)$, the remainder is $5$.

Feb/March 2021

The variables $x$ and $y$ are related by $y = 3^{2a} a^x$, where $a$ is constant. A plot of $\ln y$ against $x$ produces a straight line with gradient $0.239$.

Feb/March 2022

On one pair of axes, sketch the graphs of $y = |2x - 11|$ and $y = 3x - 3$.

Feb/March 2023

Solve the equation $3^{4x+3} = 5^{2x+7}$ by using logarithms. Give your answer correct to $3$ significant figures.

Feb/March 2024

Solve for $x$ in $\ln(3x + 1) - \ln(x - 5) = \ln 7$.

Feb/March 2025

Given that $y = 2^x$, show that $2^x + 3(2^{-x}) = 4$ can be expressed as $y^2 - 4y + 3 = 0$.

May/June 2010

Use logarithms on $13^x = (2.8)^y$ to show that $y = kx$, then determine $k$ correct to $3$ significant figures.

May/June 2010

Starting from $13^x = (2.8)^y$, apply logarithms to demonstrate that $y = kx$ and determine the value of $k$ correct to $3$ significant figures.

May/June 2010

Employ logarithms to find the solution of $3^x = 2^{x+2}$, giving the answer correct to 3 significant figures.

May/June 2011

Apply logarithms to solve the equation $3^x = 2^{x+2}$, and give the answer correct to 3 significant figures.

May/June 2011

The variables $x$ and $y$ are linked by $y = A(b^x)$, where $A$ and $b$ are constants. A graph of $\ln y$ against $x$ is a straight line that passes through $(0, 2.14)$ and $(5, 4.49)$, as shown in the diagram.

May/June 2012

If $5^{2x} + 5^x = 12$, determine the value of $5^x$.

May/June 2012

The variables $x$ and $y$ are related by the equation $y = A(b^x)$, where $A$ and $b$ are constants. The graph of $\ln y$ plotted against $x$ forms a straight line and passes through the points $(0, 2.14)$ and $(5, 4.49)$, as shown in the diagram.

May/June 2012

Solve the equation $|2^x - 7| = 1$, and give answers to $2$ decimal places when needed.

May/June 2013

Solve $\ln(3 - 2x) - 2\ln x = \ln 5$.

May/June 2013

The variables $x$ and $y$ are linked by the equation $5^{y+1} = 2^{3x}$.

May/June 2013

Solve $|2^x - 7| = 1$, with answers stated correct to $2$ decimal places where needed.

May/June 2013

Solve for $x$ in the equation $\ln(3 - 2x) - 2\ln x = \ln 5$.

May/June 2013

Solve $|x + 2| = |x - 13|$.

May/June 2014

Solve $|x + 2| = |x - 13|$.

May/June 2014

Solve the equation $|3x + 4| = |3x - 11|$.

May/June 2015

The variables $x$ and $y$ obey $y = A e^{p(x-1)}$, with $A$ and $p$ as constants. The graph of $\ln y$ plotted against $x$ is a straight line that goes through the points $(2, 1.60)$ and $(5, 2.92)$, as the diagram indicates.

May/June 2015

Apply logarithms to find the solution of the equation $2^x = 20^5$, with the final answer correct to $3$ significant figures.

May/June 2015

Solve the equation $2^x = 20^5$ by using logarithms, and give your answer correct to 3 significant figures.

May/June 2015

If $3e^x + 8e^{-x} = 14$.

May/June 2016

Because $5^{3x} = 7^{4y}$.

May/June 2016

Suppose that $5^{3x} = 7^{4y}$.

May/June 2016

Use logarithms to solve $3^{x+4} = 5^{2x}$, and give the answer correct to $3$ significant figures.

May/June 2017

Apply logarithms to solve $3^{x+4} = 5^{2x}$, and give the value of $x$ correct to $3$ significant figures.

May/June 2017

Solve the equation $3e^{2x} - 82e^{x} + 27 = 0$, and give your answers in the form $k\ln 3$.

May/June 2018

Find the solution of the equation $2\ln(2x) - \ln(x + 3) = 4\ln 2$.

May/June 2018

Find the value of $x$ in the equation $2\ln(2x) - \ln(x + 3) = 4\ln 2$.

May/June 2018

Show that the relation $\ln(x^3 - 4x) - \ln(x^2 - 2x) = \ln(x + 2)$ holds.

May/June 2019

Solve the inequality $|3x - 5| < |x + 3|$ for $x$.

May/June 2019

Find the values of $x$ that satisfy the equation $|4 + 2x| = |3 - 5x|$.

May/June 2019

Find the solutions of $|4 + 2x| = |3 - 5x|$.

May/June 2019

Solve $\ln(x + 1) - \ln x = 2 \ln 2$.

May/June 2020

With $2^y = 9^{3x}$ given, use logarithms to show that $y = kx$ and determine the value of $k$ correct to $3$ significant figures.

May/June 2020

Since $2^y = 9^{3x}$,

May/June 2020

The variables $x$ and $y$ are related by the equation $y = Ax^{-2p}$, where $A$ and $p$ are constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line passing through the points $(-0.68, 3.02)$ and $(1.07, -1.53)$, as illustrated in the diagram.

May/June 2020

The polynomial $p(x)$ is defined as $p(x) = ax^3 - 11x^2 - 19x - a$, where $a$ is a constant. It is known that $(x - 3)$ is a factor of $p(x)$.

May/June 2021

Solve $\ln(2 + x) - \ln x = 2\ln 3$.

May/June 2021

Determine the quotient when $x^4 - 32x + 55$ is divided by $(x - 2)^2$, and show that the remainder equals $7$.

May/June 2021

Solve for $x$ in the equation $\ln(2 + x) - \ln x = 2\ln 3$.

May/June 2021

Determine the quotient when $x^4 - 32x + 55$ is divided by $(x - 2)^2$ and show that the remainder is $7$.

May/June 2021

The variables $x$ and $y$ obey the equation $y = 4^{2x-a}$, where $a$ is an integer. As the diagram indicates, the graph of $\ln y$ against $x$ forms a straight line that passes through $(0, -20.8)$, with the second coordinate stated correct to $3$ significant figures.

May/June 2022

The polynomial $p(x)$ is given by $p(x) = 2x^3 + ax^2 - 3x - 4$, with $a$ a constant. It is stated that $(x - 4)$ is a factor of $p(x)$.

May/June 2022

The polynomial $p(x)$ is given by $p(x) = 2x^3 + ax^2 - 3x - 4$, with $a$ a constant. It is stated that $(x - 4)$ is a factor of $p(x)$.

May/June 2022

Use logarithms to determine the solution of the equation $12^x = 3^{2x+1}$. Give your answer correct to 3 significant figures.

May/June 2023

The variables $x$ and $y$ obey the equation $y = Ae^{(A-B)x}$, where $A$ and $B$ are constants. The graph of $\ln y$ against $x$ is a straight line that passes through $(0.4, 3.6)$ and $(2.9, 14.1)$, as illustrated in the diagram.

May/June 2023

Apply logarithms to solve the equation $6^{2x - 1} = 5e^{3x + 2}$. State your answer to $4$ significant figures.

May/June 2024

Solve $6^{2x - 1} = 5\mathrm{e}^{3x + 2}$ by using logarithms. Give your answer correct to $4$ significant figures.

May/June 2024

Solve the inequality $4^x < 0.05$ by using logarithms. Present your answer in the form $x < a$, with $a$ correct to 3 significant figures.

May/June 2025

Use logarithms to solve $5^x = 2^{2x+1}$, and give your answer correct to 3 significant figures.

Oct/Nov 2010

Use logarithms to determine the value of $x$ for $5^x = 2^{2x+1}$, with the answer given correct to $3$ significant figures.

Oct/Nov 2010

Let $x$ and $y$ be linked by $y = A(b^x)$, with $A$ and $b$ as constants. The plot of $\ln y$ against $x$ is a straight line that passes through the points $(1.4, 0.8)$ and $(2.2, 1.2)$, as the diagram shows.

Oct/Nov 2010

Solve the equation $3^{2x} - 7(3^x) + 10 = 0$, and give your answers correct to $3$ significant figures.

Oct/Nov 2011

Solve $4^{x+1} = 5^{2x-3}$ using logarithms, and give your answer correct to 3 significant figures.

Oct/Nov 2011

Solve $2\ln(x + 3) - \ln x = \ln(2x - 2)$.

Oct/Nov 2011

Use logarithms to solve the equation $5^x = 3^{2x-1}$, and give your answer correct to $3$ significant figures.

Oct/Nov 2012

The variables $x$ and $y$ are linked by $y = A b^{-x}$, with $A$ and $b$ as constants. As illustrated, the graph of $\ln y$ against $x$ is a straight line that goes through the points $(1, 2.9)$ and $(3.5, 1.4)$.

Oct/Nov 2012

Use logarithms to find the solution of the equation $5^x = 3^{2x-1}$, giving your answer correct to $3$ significant figures.

Oct/Nov 2012

Given that $(x + 2)$ and $(x + 3)$ are factors of $5x^{3} + ax^{2} + b$, determine the values of the constants $a$ and $b$.

Oct/Nov 2014

Determine the value of $x$ that satisfies the equation $2\ln(x - 4) - \ln x = \ln 2$.

Oct/Nov 2014

Since $(x + 2)$ and $(x + 3)$ are factors of $5x^3 + ax^2 + b$, determine the constants $a$ and $b$.

Oct/Nov 2014

Solve $5^{x+3} = 7^{x-1}$ by using logarithms, and give the result correct to 3 significant figures.

Oct/Nov 2015

Solve $|3x - 2| = 5$.

Oct/Nov 2015

Find the $x$-coordinates of the stationary points for the curves below:

Oct/Nov 2015

Solve for $x$ in the equation $|2x+3| = |x+8|$.

Oct/Nov 2015

The equation $3^{2x} = 5(3^x) + 14$ is satisfied by $x$.

Oct/Nov 2016

The variables $x$ and $y$ are linked by the equation $y = Ae^{px}$, where $A$ and $p$ are constants. A plot of $\ln y$ against $x$ gives a straight line that goes through the points $(5, 3.17)$ and $(10, 4.77)$, as the diagram shows.

Oct/Nov 2016

If $\frac{1 + 4y}{3 + 2^y} = 5$, determine the value of $2^y$.

Oct/Nov 2016

The polynomial $p(x)$ is given by $p(x) = ax^3 + 3x^2 + 4ax - 5$, where $a$ is a constant. It is stated that $(2x - 1)$ is a factor of $p(x)$.

Oct/Nov 2016

Solve the equation $\ln(3x + 1) - \ln(x + 2) = 1$, and give the answer in terms of $e$.

Oct/Nov 2017

Apply logarithms to solve the equation $5^{3x-1} = 2^{4x}$, and give your answer correct to $3$ significant figures.

Oct/Nov 2017

Solve the equation $\ln(3x + 1) - \ln(x + 2) = 1$, expressing your answer in terms of $e$.

Oct/Nov 2017

Solve for $x$ in the equation $\lvert 9x - 2 \rvert = \lvert 3x + 2 \rvert$.

Oct/Nov 2018

If $9^x + 3^x = 240$, determine the value of $3^x$ and then, by using logarithms, find $x$ correct to $4$ significant figures.

Oct/Nov 2018

Solve for $x$ in $|9x - 2| = |3x + 2|$.

Oct/Nov 2018

The polynomial $p(x)$ is given by $p(x)=ax^3+ax^2-15x-18$, where $a$ is a constant. It is stated that $(x-2)$ is a factor of $p(x)$.

Oct/Nov 2019

Solve $|4x + 5| = |x - 7|$.

Oct/Nov 2019

The polynomial $p(x)$ is given by $p(x) = ax^3 + ax^2 - 15x - 18$, with $a$ a constant. It is stated that $(x - 2)$ is a factor of $p(x)$.

Oct/Nov 2019

Assume that $\ln(2x + 1) - \ln(x - 3) = 2$.

Oct/Nov 2020

Solve for $x$ in $|2x - 5| = |x + 6|$.

Oct/Nov 2020

Starting from $\frac{2^{3x+2} + 8}{2^{3x} - 7} = 5$, determine $2^{3x}$ and then, by taking logarithms, determine $x$ correct to $4$ significant figures.

Oct/Nov 2020

The equation $\ln(2x + 1) - \ln(x - 3) = 2$ is given.

Oct/Nov 2020

On one set of axes, sketch the graphs of $y = x + 3$ and $y = |2x - 1|$.

Oct/Nov 2021

Use logarithms to find the solution of $14e^{-2x} = 5^{x+1}$, giving your answer correct to $3$ significant figures.

Oct/Nov 2022

The polynomial $p(x)$ is defined by $p(x) = ax^3 + 23x^2 - ax - 8$, where $a$ is a constant. You are told that $(2x + 1)$ is one factor of $p(x)$.

Oct/Nov 2022

Use logarithms to solve the equation $14e^{-2x} = 5^{x + 1}$, and give your answer correct to $3$ significant figures.

Oct/Nov 2022

On one set of axes, sketch the graphs of $y = |3x - 5|$ and $y = 2x + 7$.

Oct/Nov 2023

Sketch, on one set of axes, the graphs of $y = |3 - x|$ and $y = 9 - 2x$.

Oct/Nov 2023

On one set of axes, sketch the graphs of $y = |3x - 5|$ and $y = 2x + 7$.

Oct/Nov 2023

The variables $x$ and $y$ are related by $a^{2y} = e^{3x+k}$, with $a$ and $k$ constant. A graph of $y$ against $x$ is a straight line.

Oct/Nov 2024

Use logarithms to show that $5^{8y} = 6^{7x}$ may be written in the form $y = kx$. State the value of the constant $k$ correct to 3 significant figures.

Oct/Nov 2024