Logarithmic & Exponential Functions (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

On the same axes, sketch the graphs of $y=2^x$ and $y=3^x$, labelling any points where the graphs cross the coordinate axes and writing down the equation of any asymptotes.

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

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

Using a calculator, find to 3 significant figures

(i) ${\text{log}}_{2}5+{\text{log}}_{5}2$

(ii) $\text{log}25-\text{ln}2$

(iii) $\text{log}200+{\text{log}}_{5}50-\text{log}20+\text{ln}10$

Solve $3 {\text{log}}_{2}4+3x=5 {\text{log}}_{6}216$.

Solve $2^{2x}-24\left(2^x\right)+128=0$.

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 $(0,1)$.

(i) Write down the gradient function 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 of $x$ for which the gradient of $y=e^{-3x}$ is $-3e^{-12}$.

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

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

(ii) On the same axes, sketch the graphs of $y=f\left(x\right)$ and $y=f\left(-x\right)$. Label any points where the graphs intersect the coordinate axes.

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

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

On the same axes, sketch the graphs of $y=4^x$ and $y=5^x$.

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

Write down an equation for the graph that is a reflection of $y=4^x$ in the $y$-axis.

(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 $y=0.064$.

Find the value of $\text{log}1000+\text{log}10000$.

Write down the value of $a$ in the statement $6^{{\text{log}}_{6}a}=36$.

Evaluate $\frac{2 {\text{log}}_{4}64+3^{{\text{log}}_{2}8}-{\text{log}}_{5}5}{\text{log}100}$.

Solve $2 \text{log}1000=x {\text{log}}_{16}4$.

Solve $3 {\text{log}}_{4}x={\text{log}}_{4}x+3 {\text{log}}_{5}25$.

Solve $2\left(2^{2x}\right)+4=9\left(2^x\right)$.

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

Find $f\left(2x\right)$.

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

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

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

On the same axes, sketch the graphs of $y=0.3^x$ and $y=0.5^x$.

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

Write down an equation for the graph that is a reflection of $y=0.5^x$ in the $y$-axis.

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

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

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

Find $f\left(4x\right)$.

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

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 where $x=0$ the gradient is 10. Find $y$ 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.

Without using a calculator, evaluate ${\text{log}}_{4}128$.

Show each stage of your solution carefully.

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

Solve $3^{2\left(x+1\right)}+3=28\left(3^x\right)$.

Find two values of $x$ for which $\text{log}\left(x^2\right)={\left(\text{log}x\right)}^2$ is true.

A particle is travelling with velocity $v{\text{m s}}^{-1}$ at time $t$ seconds. The velocity of the particle is modelled by $v=0.3e^{kt}$, 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 $0.9{\text{m s}}^{-1}$.

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.