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

Sketch the graph of , stating whether this graph indicates exponential growth or exponential decay.

Find the exact value of when .

Write down the value of in the following statements.

On the same axes, sketch the graphs of and .
Label any points where the graphs intersect the coordinate axes.
Write down the equations of any asymptotes.

Write down the gradient of at the point .

Given write down an expression for .

Given write down an expression for .

Find the gradient of at the point where .
Give your answer in the form , where  and  are integers to be found.

(i) Write down the gradient of .

(ii) Find the gradient of  at the point where .

(i)   In terms of , write down the gradient of at the point where .

(ii)  Find the value for  for which the gradient of  is .

The function  is defined by  for

Find .

On the same axes, sketch the graphs of  and .

          Label any points where the graphs intersect the coordinate axes.

Describe the transformation from to .

Solve , giving your answers to 3 significant figures.

Evaluate 

Evaluate

giving your answer in the form  where  is an integer to be found.

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

Write the following as a single algorithm

Write the following in the form where  and  are integers to be found.

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, where  is an integer to be found.

Show that

Sketch the graph of .

Find the value of  when .

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

Write down the gradient of at the point where .

The function  is defined by  for .

Find .

Find .

Solve , giving your answer to 3 significant figures.

Find the gradient of the curve ,  where and and  are constants.

At the point the gradient is 12, find in terms of .

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

Show that the equation has only one real solution.

Explain why the equation has no real solutions.

Evaluate

Evaluate

giving your answer in the form 

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

Simplify

giving your answer in the form , where  and  are integers to be found.

Write

as a single algorithm.

Solve the equations

giving your answer in the form  where  and  are integers to be found.

A ship sets from a harbour.

After some time, the ship's position is  east of the harbour and   north of the harbour.

Find the direct distance between the ship and the harbour at this time giving your answer in the form .

By writing  as show that

can be written as .

Sketch the graph of , stating whether this graph indicates exponential growth or exponential decay.

Find the value of when .

Without using a calculator, evaluate .
Show each stage of your solution carefully.

Evaluate .

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

Find the gradient of at the point where , giving your answer correct to 3 significant figures.

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

.
Write down the initial population.

The function  is defined by  for .

Find .

Find .

Find the gradient of the curve where and  are constants.

State a condition on to ensure represents exponential decay.

At the point the gradient is 10. Find  in terms of (and ) only.

A particle is travelling with velocity,  ms^-1, at time  seconds.
The velocity of the particle is modelled , where  is a constant.

Write down the initial velocity of the particle.

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

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

State a problem with the model for large values of 

Solve .

Solve the equation , giving answers to 3 significant figures where appropriate.

Evaluate

Evaluate

giving your answer in the form .

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

On the same axes, sketch the graphs of  and 

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 

Show that  can be written in the form ().

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 .

The two smallest angles in the triangle are  and respectively where  .

Find all three sides of the triangle, giving your answers in the form 

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