Exponential & Logarithms (OCR A Level Maths: Pure)

On the same axes, sketch the graphs of  and , labelling any points where the graphs cross the coordinate axes and writing down the equation of any asymptotes.

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.

Using a calculator, find to 3 significant figures

Solve .

Solve .

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.

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.

On the same axes, sketch the graphs of and .

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 in the -axis.

Sketch the graph of .

Find the value of  when .

Find the value of 

Write down the value of in the statement .

Evaluate .

Solve .

Solve .

Solve .

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.

On the same axes, sketch the graphs of and .

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 in the -axis.

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 .

Solve .

Find two values of  for which  is true.

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, P, 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.