Laws of logarithms

Given that

find the exact value of

Figure 1 shows a sketch of the curves with equations y = f(x) and y = g(x) where

Find

f′(x)

g′(x)

Given that f′(x) = g′(x) at x = α

show that α satisfies the equation

The iterative formula

is used withx₁ = 0.6 to find an approximate value for α

Calculate, giving each answer to 4 decimal places,

the value of x₂

the value of α

a = log₂ x b = log₂(x + 8) Express in terms of a and/or b

log₂(x² + 8x)

Give your answer in simplest form.

Given that log₂(x + 3) + log₂(x + 10) = 2 + 2 log₂ x

show that

Write down the roots of the equation

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

giving a reason for your answer.

A scientist is studying the growth of two different populations of bacteria. The number of bacteria, N, in the first population is modelled by the equation

where A and k are positive constants and t is the time in hours from the start of the study. Given that

• there were 1000 bacteria in this population at the start of the study

• it took exactly 5 hours from the start of the study for this population to double

find a complete equation for the model.

Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. The number of bacteria, M, in the second population is modelled by the equation

where k has the value found in part (a) and t is the time in hours from the start of the study. Given that T hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of T.

Using the laws of logarithms, solve the equation

Using the laws of logarithms, solve the equation

The time, T seconds, that a pendulum takes to complete one swing is modelled by the formula

where l metres is the length of the pendulum and a and b are constants.

Show that this relationship can be written in the form

A student carried out an experiment to find the values of the constants a and b.

The student recorded the value of T for different values of l.

Figure 3 shows the linear relationship between log₁₀ l and log₁₀ T for the student’s data. The straight line passes through the points (– 0.7, 0) and (0.21, 0.45)

Using this information,

find a complete equation for the model in the form

giving the value of a and the value of b, each to 3 significant figures.

With reference to the model, interpret the value of the constant a.

Given that 2 log (4 − x) = log (x + 8) show that

Write down the roots of the equation

State which of the roots in (b)(i) is not a solution of 2 log (4 − x) = log (x + 8) giving a reason for your answer.

Given that a b 0 and that a and b satisfy the equation

log a – log b = log(a – b)

show that

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

Find the value of y for which

Give your answer to 2 decimal places.

Given that

show that

Hence solve the equation

y = 5^x + log₂(x + 1), 0 x ≤ 2 Complete the table below, by giving the value of y when x = 1

Use the trapezium rule, with all the values of y from the completed table, to find an approximate value for

giving your answer to 2 decimal places.

Use your answer to part (b) to find an approximate value for

giving your answer to 2 decimal places.

2 log(x + a) = log(16a⁶), where a is a positive constant Find x in terms of a, giving your answer in its simplest form.

log₃(9y + b) – log₃(2y – b) = 2, where b is a positive constant Find y in terms of b, giving your answer in its simplest form.

Given that log₃(3b + 1) – log₃(a – 2) = –1, a 2 express b in terms of a.

Solve the equation

giving your answer to 2 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Given that log₃ x = a, find in terms of a,

giving each answer in its simplest form.

Solve, for x,

giving your answer to 4 significant figures.

Find the exact value of x for which

Given that

express y in terms of a. Give your answer in its simplest form.

Given that

Show that

Hence, or otherwise, solve the equation

Find the values of x such that

Find, giving your answer to 3 significant figures where appropriate, the value of x for which

Find the value of y such that

Find the values of x such that