Solving exponential equations

Sketch the curve with equation

stating any points of intersection with the coordinate axes.

Solve

giving your answer to 2 decimal places.

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.

By taking logarithms of both sides, solve the equation

giving the value of p to one decimal place.

The curve with equation y = 3 × 2^x meets the curve with equation y = 15 − 2^x+1 at the point P. Find, using algebra, the exact x coordinate of P.

The value, £ V, of a vintage car t years after it was first valued on 1st January 2001, is modelled by the equation

V=Ap^twhere A and p are constants

Given that the value of the car was £32 000 on 1st January 2005 and £50 000 on 1st January 2012

find p to 4 decimal places,

show that A is approximately 24 800

With reference to the model, interpret

the value of the constant A,

the value of the constant p.

Using the model,

find the year during which the value of the car first exceeds £100 000

A scientist is studying a population of mice on an island.

The number of mice, N, in the population, t months after the start of the study, is modelled by the equation

Find the number of mice in the population at the start of the study.

The rate of growth is a maximum after T months.

Find, according to the model, the value of T.

According to the model, the maximum number of mice on the island is P.

State the value of P.

Find the value of y for which

Give your answer to 2 decimal places.

Given that

show that

Hence solve the equation

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.)

Sketch the graph of

showing the coordinates of any points at which the graph crosses the axes.

Use algebra to solve the equation

giving your answers to 2 decimal places where appropriate.

f(x) = 2x³ – 5x² + ax + 18 where a is a constant. Given that (x – 3) is a factor of f(x),

show that a = – 9

factorise f(x) completely.

Given that

find the values of y that satisfy g(y) = 0, giving your answers to 2 decimal places where appropriate.

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.

A company predicts a yearly profit of £120 000 in the year 2013. The company predicts that the yearly profit will rise each year by 5%. The predicted yearly profit forms a geometric sequence with common ratio 1.05

Show that the predicted profit in the year 2016 is £138 915

Find the first year in which the yearly predicted profit exceeds £200 000

Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound.

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

Find, to 3 significant figures, the value of x for which 5^x = 7.

Solve the equation.5^2x 12(5^x) 35 0