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.

Express

in the form

where and is an expression in terms of , and that you should find.

The diagram below shows the length of three sides of a triangle.

Find the perimeter of the triangle, giving your answer in the form

where  is an integer to be found.

Simplify fully the expression

Solve the equation

Solve the equation

giving your answer to 3 significant figures.

(i) On the same set of coordinate axes, sketch the curves and .

Label clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) State the equation of any asymptotes and the equation of the line of reflection between the two curves.

A ship sets sail from a harbour.

After some time, the ship’s position is  km due east of the harbour and  km due north of the harbour.

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

Express

in the form where is an irrational number to be found.

Solve

giving your answer(s) to 3 significant figures.

Write

in the form  where is a constant to be found.

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

Solve the equation

giving your answer(s) in the form where is a constant to be found.

Express

in the form

where is a function that you should state.

Solve the equation

giving your answer(s) in the form where is a constant to be found.

Solve the equation

giving your answer in the form

where  and  are positive integers to be found.

Using the laws of logarithms, solve the equation

giving your answer(s) to 3 significant figures.

Solve the equation

giving your answer(s) in the form , where is a constant to be found.

Show that

can be expressed in the form

where .

Express

in the form

where , and are integers to be found.

Given that , write each of the following in terms of

(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.