Logarithms

The function g is defined by

where k is a constant.

Deduce the value of k.

Prove that

for all values of x in the domain of g.

Find the range of values of a for which

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 exact solutions, in their simplest form, to the equations

2 ln(2 x + 1) – 10 = 0

The function f is defined by

State the range of f.

Find f^–1 and state its domain.

The function g is defined by

Solve the equation

giving your answer in its simplest form.

Find fg(x), giving your answer in its simplest form.

Find, in terms of the constant k, the solution of the equation

A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation

Calculate the number of primroses at the start of the study.

Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers.

Find the exact value of when t =10. Give your answer in its simplest form.

Explain why the population of primroses can never be 270

Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of

y = f(x),

y = | f(x) |,

y = –f(x – 4). Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote.

Find algebraically the exact solutions to the equations

ln(4 – 2 x) + ln(9 – 3 x) = 2ln(x + 1), –1 < x < 2

2^x e^3x+1 = 10 Give your answer to (b) in the form where a, b, c and d are integers.

The functions f and g are defined by

State the range of f.

Find fg(x), giving your answer in its simplest form.

Find the exact value of x for which f (2x 3) 6

Find f ¹, the inverse function of f, stating its domain.

On the same axes sketch the curves with equation y = f (x) and y = f ¹(x), giving the coordinates of all the points where the curves cross the axes.

The function f is defined by

Find the domain of f ¹.

The function g is defined by

Find fg (x), giving your answer in its simplest form.

Find the range of fg.

Simplify fully

Given that ln(2x² + 9x− 5) = 1+ ln(x² + 2x 15), x 5,

find x in terms of e.

Sketch the graph of y = ln x, stating the coordinates of any points of intersection with the axes.

Find the exact solutions to the equations

ln (3 x – 7) = 5

The functions f and g are defined by

Find f^–1 and state its domain.

Find fg and state its range.

The curve meets the coordinate axes at the points A(0,1–k) and B (–¹₂ k ln,0), where k is a constant and k 1, as shown in Figure 2. On separate diagrams, sketch the curve with equation

y = f ¹(x).

Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.

Given that f (x) = e^2x − k,

state the range of f,

write down the domain of f ¹.

The function f is defined by

The function g is defined by

Find the exact values of x for which g ( x) = 1