Logarithmic & Exponential Functions (Cambridge International A-Level Mathematics): Exam Questions

Sketch the graph with equation $y = a^{x}$, $a > 1$, stating the coordinates of the point where the graph intersects the $y$-axis and the equation of any asymptotes.

Also state whether this equation would represent exponential growth or decay.

The following equations can be used for exponential models.

State whether each one would represent exponential growth or exponential decay.

(i) $y = 3^{-2x}$

(ii) y = 20 \left(2\right)^{x}

(iii) $y = 30 a^{-x}$ where $a > 1$

Write down the value(s) of $a$ in the following statements:

(i) $3^{a} = 27$

(ii) $a^{\frac{1}{3}} = 5$

(iii) $4 a^{2} = 64$

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

(i) $\text{log}_{3} a = 4$

(ii) $\text{log}_{a} 216 = 3$

(iii) $\text{log}_{2} 128 = a$

Solve the equation $2^{x} = 16$.

Solve the equation $x^{2} - 12 x + 27 = 0$.

Hence, or otherwise, solve the equation \left(3^{x}\right)^{2} - 12\left(3^{x}\right) + 27 = 0.

Solve the equation $2 \text{log}_{3} 9 = 5 x - 6$.

Sketch the graph of $y = e^{x}$, clearly showing the coordinates of the point where the graph intercepts the $y$-axis and stating the equations of any asymptotes.

Given $y = e^{2x}$:

(i) Write down an expression for $\frac{dy}{dx}$.

(ii) Find the gradient of $y = e^{2x}$ at the point where $x = 0$.

Use a calculator to find the value of

(i) $5 \text{log}_{3} 7$

(ii) $2 \text{log}_{2} 3 + 3 \text{log}_{3} 2$

giving your answers to four significant figures.

Solve the equation $e^{2x} - 16 = 0$, giving your answer in the form $a \text{ln} a$ where $a$ is an integer.

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

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

(ii) $\text{log} a = 2$

(iii) $\text{ln} e^{3} = a$

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

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

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

Find the gradient of $y = 3 e^{-2x}$ at the point where $x = 3$.

Give your answer in the form $p e^{q}$, where $p$ and $q$ are integers to be found.

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

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 $- 3 e^{-12}$.

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

(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 $e^{2x} - 8 e^{x} + 15 = 0$, 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 $y = 12 e^{-x}$ for $x \geq 0$.

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

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

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

Find f\left(2x\right).

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

Solve $2 e^{2x} = e^{x} + 10$, 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 $y = 4 e^{x}$ for $x \geq 0$.

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

Find the gradient of $y = 4 e^{x}$ 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 $P = 4 e^{t}$.

Write down the initial population.

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

Find f\left(4x\right).

Find f'\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 $v = 0 . 3 e^{kt}$, 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 $2 e^{3x} - 11 e^{2x} + 12 e^{x} = 0$, giving answers to 3 significant figures where appropriate.