Logarithmic & Exponential Functions (Cambridge (CIE) AS Maths: Pure 2): 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 $x=3$.

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

Using a calculator, find to 3 significant figures

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

(ii) $\log 25-\ln 2$

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

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

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 $\left(0,1\right)$.

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

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

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 function of $y={\mathrm{e}}^{-3x}$

(ii) Find the gradient of $y={\mathrm{e}}^{-3x}$ 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.

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 $\log 1000+\log 10000$

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

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

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

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

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 $\mathrm{f}\left(x\right)$ 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.

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 $y=625$.

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

Show each stage of your solution carefully.

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

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

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

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)$$=5e^{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.