Laws of Logarithms (Edexcel A Level Maths: Pure): Exam Questions

Given that

show that

(i) Write down the roots of the equation

(ii) State which of the roots in (b)(i) is not a solution of

giving a reason for your answer.

Using the laws of logarithms, solve the equation

Given that and that and satisfy the equation

show that

Write down the full restriction on the value of , explaining the reason for this restriction.

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.

By writing , show that

Write the following as a single logarithm

Write the following in the form , where a and b 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

Solve the equation

Solve the equation

(i) On the same axes, sketch the graphs of  and .
Label any points of intersection between each graph and the coordinate axes.
Write down the equations of any asymptotes.

(ii) Write down the equation of the line of reflection between the graphs of  and   

Solve the equation

Given

express as a function of .

By taking logarithms of both sides, solve the equation

giving the value of to one decimal place.

Given that , where , find in simplest form in terms of ,

(i)

(ii)

Hence, or otherwise, solve

giving your answer as a simplified fraction.

Solutions relying on calculator technology are not acceptable.

Using the laws of logarithms, solve the equation

Evaluate

Evaluate

giving your answer in the form .

Solve the equation

Solve the equation

giving your answer to 3 significant figures.

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 logarithm.

(i) On the same axes, sketch the graphs of and  On each graph, label any points where the graph intersects the coordinate axes.

Write down the equations of any asymptotes for each graph.

(ii) Write down the line of reflection between the graphs  and .

Solve the equation

giving your answers in the form .

Solve the equation

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

A ship sets sail 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 km.

By writing 5 as , show that 

can be written as 

Solve the equation

giving your answers correct to 3 significant figures.

Solve the equation

giving your answer correct to 3 significant figures.

Using your answer to part (a) and making your method clear, estimate

(i)

(ii)

Given that

show that

(i) Write down the roots of the equation

(ii) Hence state which of the roots in part (b)(i) is not a solution of

giving a reason for your answer.

The curve with equation meets the curve with equation at the point .

Find, using algebra, the exact coordinate of .

Express in terms of and/or

Give your answer in simplest form.

Evaluate

Evaluate

giving your answer in the form .

Solve the equation

giving your answers in the form  , where and  are rational numbers to be found.

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

Show that

Write the following as a single logarithm

On the same axes, sketch the graphs of  and

On each graph, label any points where the graph 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 

Α 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 cm.

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 equation have?  Justify your answer.

Without using a calculator, show that