Exponential & Logarithms (AQA AS Maths: Pure): Exam Questions

On the same set of coordinate axes, sketch the curves

$y=2^x$

and

$y=3^x$

Label clearly

• any points where the curves meet the coordinate axes

• the equation(s) of any asymptotes

Sketch the curve

$y=\frac{1}{2^x}$

State whether the graph shows exponential growth or exponential decay.

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

Solve the equation

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

Use algebra to solve

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

On the same set of coordinate axes, sketch the curves

$y={\mathrm{e}}^x$

and

$y={\mathrm{e}}^{-x}$

Label clearly

• any points where the curves meet the coordinate axes

• the equation(s) of any asymptotes

Write down the gradient of the curve

$y={\mathrm{e}}^x$

at the point $\left(0,1\right)$.

Given that

$y={\mathrm{e}}^{4x}$

write down an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$

Given that

$y=2{\mathrm{e}}^{2x}$

write down an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$

A curve has equation

$y=3{\mathrm{e}}^{-2x}$

Find the gradient of the point on the curve where $x=3$.

Give your answer in the form

$p{\mathrm{e}}^q$

where $p$ and $q$ are integers to be found.

(i) Write down the gradient function of the curve $y={\mathrm{e}}^{-3x}$

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

(i) write down the exact gradient of the point on the curve $y={\mathrm{e}}^{-3x}$ where $x=2$

(ii) Find the $x$-coordinate of the point on the curve $y={\mathrm{e}}^{-3x}$ which has a gradient of $-3{\mathrm{e}}^{-12}$

The function $\mathrm{f}\left(x\right)$ is defined by

$\mathrm{f}\left(x\right)=2{\mathrm{e}}^{3x}x\in \mathrm{ℝ}$

(i) Write down the function $\mathrm{f}\left(-x\right)$.

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

Label any points where the graphs intersect the coordinate axes.

A different function is given by

$\mathrm{g}\left(x\right)=-2{\mathrm{e}}^{3x}$

State the single graph transformation that maps the curve $y=\mathrm{f}\left(x\right)$ to the curve $y=\mathrm{g}\left(x\right)$.

Solve

${\mathrm{e}}^{2x}-8{\mathrm{e}}^x+15=0$

giving your answer(s) to 3 significant figures.

On the same set of coordinate axes, sketch the curves

$y=4^x$

and

$y=5^x$

Label clearly

• any points where the curves meet the coordinate axes

• the equation(s) of any asymptotes

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

(i) Sketch the curve with equation $y=0.4^x$

(ii) State whether the curve represents exponential growth or exponential decay.

Find the value of $x$ when $y=0.064$.

Find the value of

$\log 1000+\log 10000$

Solve

$6^{{\log }_{6}a}=36$

Find the value of

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

Sketch the curve with equation

$y=12{\mathrm{e}}^{-x}x\ge 0$

stating any points of intersection with the coordinate axes.

Write down the equations of any asymptotes.

Find the gradient of the point on the curve $y=12{\mathrm{e}}^{-x}$ where $x=0$.

Solve

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

Solve

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

On the same set of coordinate axes, sketch the curves

$y=0.3^x$

and

$y=0.5^x$

Label clearly

• any points where the curves meet the coordinate axes

• the equation(s) of any asymptotes

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

Sketch the curve

$y=0.2^{-x}$

State

• the equation of the asymptote

• whether the graph shows exponential growth or exponential decay.

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

Sketch the curve with equation

$y=4{\mathrm{e}}^{x}x\ge 0$

stating any points of intersection with the coordinate axes.

Write down the equations of any asymptotes.

Find the gradient of the point on the curve $y=4{\mathrm{e}}^x$ where $x=3$.

Give your answer correct to 3 significant figures.

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

$P=4{\mathrm{e}}^t$

Write down the initial population.

Solve

$2{\mathrm{e}}^{2x}={\mathrm{e}}^x+10$

giving your answer(s) to 3 significant figures.

Solve

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

Show that the equation

${\mathrm{e}}^x-{\mathrm{e}}^{-x}=0$

has only one real solution, which you should find.

Show that the equation

${\mathrm{e}}^x+{\mathrm{e}}^{-x}=0$

has no real solutions.

Without using a calculator, find the value of

$\frac{3{\log }_{6}216-\ln \left({\mathrm{e}}^5\right)+4^{{\log }_{5}625}}{\log 10000}$

Show the steps in your solution clearly.

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.

A curve has equation

$y=\frac{1}{a}{\mathrm{e}}^{-bx}$

where $a$ and $b$ are non-zero constants.

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

At the point $\left(0,a\right)$ on the curve, the gradient is 10.

Use this information to find an expression for $y$ in terms of $a$ and $x$ only.