Exponential & Logarithms (Edexcel International A Level (IAL) Maths: Pure 3): Exam Questions

Sketch the graph of $y=2^{-x}$, stating whether this graph indicates exponential growth or exponential decay.

Find the exact value of $y$when $x=3$.

Write down the value of $a$ in the following statements.

(i) ${\log }_{a}8=3$

(ii) ${\log }_a=2$

(iii) $\ln e^3=a$

(iv) ${\log }_{5}a=1$

On the same axes, sketch the graphs of $y=e^x$ and $y=e^{-x}$. Label any points where the graphs intersect the coordinate axes.

Write down the equations of any asymptotes.

Write down the gradient of $y=e^x$ at the point $\left(0,1\right)$.

Given $y=e^{4x}$ write down an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Given $y=2e^{2x}$ write down an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Find the gradient $y=3e^{-2x}$ of at the point where $x=3$. Give your answer in the form $p{e}^q$, where $p$ and $q$ are integers to be found.

(i) Write down the gradient of $y=e^{-3x}$.

(ii) Find the gradient of $y=e^{-3x}$ at the point where $x=0$.

(i)   In terms of $e$, write down the gradient of $y=e^{-3x}$ at the point where $x=2$.

(ii)  Find the value for $x$ for which the gradient of $y=e^{-3x}$ is $-3e^{-12}$.

The function $\mathrm{f}\left(x\right)$ is defined by $\mathrm{f}\left(x\right)$$=2e^{3x}$ for $x\in \mathrm{ℝ}$

(i) Find $\mathrm{f}\left(-x\right)$.

(ii) On the same axes, sketch the graphs of $y=\mathrm{f}\left(x\right)$ and $y=\mathrm{f}\left(-x\right)$.

          Label any points where the graphs intersect the coordinate axes.

Describe the transformation from $y=\mathrm{f}\left(x\right)$ to $y=-\mathrm{f}\left(x\right)$.

Solve $e^{2x}-8e^x+15=0$, giving your answers to 3 significant figures.

Evaluate 

      ${\log }_{2}4+{\log }_{3}27-{\log }_{4}4$

Evaluate

      $3\mathrm{In}2+\frac{1}{2}\mathrm{In}81-2\mathrm{In}3,$

giving your answer in the form $\mathrm{In}q$ where $q$ is an integer to be found.

Solve the following equations, giving your answers in exact form.

   $e^x=5$

$3e^{2x}=9$

$e^{2x-1}=4$

Write the following as a single algorithm

            $2{\log }_{a}6+3{\log }_{a}2-{\log }_{a}4$

Write the following in the form $a\mathrm{In}b,$ where $a$ and $b$ are integers to be found.

         $2\mathrm{In}{3}^4+\mathrm{In}{3}^3-\mathrm{In}9$

The diagram below shows the length of three sides of a triangle, with each side measured in centimetres.

Work out the perimeter of the triangle, giving your answer in the form 2$\ln b$, where $b$ is an integer to be found.

Show that

      $2\mathrm{In}{x}^3-3\mathrm{In}{x}^2=0.$

(i) On the same axes, sketch the graphs of $y=e^x$ and $y=e^{-x}$.

Label any points of intersection between each graph and the coordinates axes.

Write down the equations of any asymptotes.

(ii) Write down the equation of the line of reflection between the graphs of $y=e^x$ and $y=e^{-x}$.

(i) Sketch the graph of $y=0.4^x$.

(ii) State whether this graph indicates exponential growth or exponential decay.

Find the value of $x$ when $y=0.064$.

Sketch the graph of $y=12e^{-x}$ for $x\ge 0$. Label any points of intersection with the coordinate axes. Write down the equations of any asymptotes.

Write down the gradient of $y=12e^{-x}$ at the point where $x=0$.

The function $\mathrm{f}\left(x\right)$ is defined by $\mathrm{f}\left(x\right)=3{\mathrm{e}}^{2\mathrm{x}}$ for $x\in \mathrm{ℝ}$.

Find $\mathrm{f}\left(\mathit{2}x\right)$.

Find $\mathrm{f}\text{'}\left(\mathit{2}x\right)$.

Solve $2e^{2x}=e^x+10$, giving your answer to 3 significant figures.

Find the gradient of the curve $y=a{e}^{bx}$,  where and $a$ and $b$ are constants.

At the point $\left(0,a\right)$ the gradient is 12, find $b$in terms of $a$.

Hence write down y in terms of $a$ (and $x$) only.

Show that the equation $e^x-e^{-x}=0$ has only one real solution.

Explain why the equation $e^x+e^{-x}=0$ has no real solutions.

Evaluate

      ${\log }_2{8}^2+3{\log }_{2}16-2{\log }_2{2}^5.$

Evaluate

      $3\mathrm{In}2+2\mathrm{In}5-\frac{1}{2}\mathrm{In}10000,$

giving your answer in the form $\mathrm{In}p.$

Solve the following equations, giving your answers in exact form.

   $4e^{3x-2}=12$

$3e^{2x}+8=14e^x$

Simplify

      $2\mathrm{In}{3}^4+\mathrm{In}{3}^3-\mathrm{In}9,$

giving your answer in the form $a\mathrm{In}b$, where $a$ and $b$ are integers to be found.

Write

      $2{\log }_{a}x+3{\log }_a\left(x+1\right)-{\log }_{a}4\left(x+2\right)$

as a single algorithm.

(i) On the same axes, sketch the graphs of $y=e^x$ and $y=\mathrm{In}x.$

On each graph, label any points where the graph intersects the coordinate axes.

Write down the equations of any asymptotes for each graph.

(ii) Write down the line of reflection between the graphs $y=e^x$ and $y=\mathrm{In}x.$

Solve the equations

         $6\times 3^{x-1}=6^{2x},$

giving your answer in the form $\frac{\mathrm{In}a}{\mathrm{In}b\text{'}}$ where $a$ and $b$ are integers to be found.

A ship sets from a harbour.

After some time, the ship's position is $\left(4\mathrm{In}3\right)$ $\mathrm{km}$ east of the harbour and $\left(3\mathrm{In}3\right)$ $\mathrm{km}$ north of the harbour.

Find the direct distance between the ship and the harbour at this time giving your answer in the form $\left(p\mathrm{In}3\right)\mathrm{km}$.

By writing $5$ as $5\mathrm{In}e,$ show that

         $5\mathrm{In}2+5$

can be written as $5\mathrm{In}2e$.

Sketch the graph of $y=0.2^{-x}$, stating whether this graph indicates exponential growth or exponential decay.

Find the value of $x$ when $y=625$.

Without using a calculator, evaluate ${\log }_{4}128$. Show each stage of your solution carefully.

Evaluate $\frac{3{\log }_{6}216-\ln e^5+4^{{\log }_{5}625}}{\log 10000}$.

Sketch the graph of $y=4e^x$ for $x\ge 0$.

Label any points of intersection with the coordinate axes. Write down the equations of any asymptotes.

Find the gradient of $y=4e^x$ at the point where $x=3$, giving your answer correct to 3 significant figures.

The population growth of population, $P$, at time, t years, is modelled by the equation .

$P=4e^t$.
Write down the initial population.

The function $\mathrm{f}\left(x\right)$ is defined by $\mathrm{f}\left(x\right)$$=5{\mathrm{e}}^{3x}$ for $x\in \mathrm{ℝ}$.

Find $\mathrm{f}\left(4x\right)$.

Find $\mathrm{f}\text{'}\left(5x\right)$.

Find the gradient of the curve $y=\frac{1}{a}{e}^{-bx}$ where $a$ and $b$ are constants.

State a condition on $b$ to ensure $y$represents exponential decay.

At the point $\left(0,a\right)$ the gradient is 10. Find $y$ in terms of $a$ (and $x$) only.

A particle is travelling with velocity, $v$ ms^-1, at time $t$ seconds.
The velocity of the particle is modelled $v=0.3e^{kt}$, where $k$ is a constant.

Write down the initial velocity of the particle.

Find an expression (in terms of $k$ and $t$) for the acceleration of the particle.

After 12 seconds the velocity of the particle is 0.9 ms^-1. Find the value of $k$, giving your answer to 3 significant figures.

State a problem with the model for large values of $t$

Solve ${\left(e^x-e^{-x}\right)}^2=0$.

Solve the equation $2e^{3x}-11e^{2x}+12e^x=0$, giving answers to 3 significant figures where appropriate.

Evaluate

      $4{\log }_{3}729+3{\log }_2{64}^2-3\log 100+\mathrm{In}{e}^6$

Evaluate

      $\frac{1}{2}\mathrm{In}196+\frac{1}{3}\mathrm{In}125+\frac{1}{4}\mathrm{In}81+\frac{1}{5}\mathrm{In}32$

giving your answer in the form $\mathrm{In}q$.

Solve the following equations, giving your answers correct to 3 significant figures.

$8e^{3x^{2-1}}=12$

$e^{3x}-42=2e^x\left(6e^x-7\right)$

On the same axes, sketch the graphs of $y=e^{2x}$ and $y=\frac{1}{2}\mathrm{In}x.$

On each graph, label any point where intersects the coordinate axes.

Write down the equations of any asymptotes for each graph.

Explain the significance of the line $y=x.$

Show that $4-\mathrm{In}16$ can be written in the form $4\mathrm{In}$($\frac{e}{2}$).

A triangle is drawn inside a circle such that one side of the triangle is the diameter and all three vertices of the triangle lie on the circumference.

The radius of the circle is $\left(3\mathrm{In}2\right)\mathrm{cm}$.

The two smallest angles in the triangle are $\alpha$ and $\beta$ respectively where  $\beta =2\alpha$.

Find all three sides of the triangle, giving your answers in the form $a\mathrm{In}2.$

How many real solutions does the equations have? Justify your answer.

            $3{\log }_x\left(x+1\right)=\mathrm{In}{e}^3$