Solving exponential equations

In this question you must show detailed reasoning.

Solve the following equations.

e^3y = 2

x⁴ - 3x² - 4 = 0

Sketch the graph of y = | 2 x - 5 | on the grid provided in the Printed Answer Booklet.

Solve the inequality | 2 x - 5 | < 1.

In this question you must show detailed reasoning.

Hence find all the possible integer values N that satisfy the inequality

Show that + can be written in the form , where a, b and c are

Hence solve the equation + = 2.

In this question you must show detailed reasoning.

Solve the equation 2^2y - 7 × 2^y - 8 = 0.

The owners of an online shop believe that their sales can be modelled by S = ab^t, where a and b are both positive constants, S is the number of items sold in a month and t is the number of complete months since starting their online shop.

The sales for the first six months are recorded, and the values of log₁₀S are plotted against t in the graph below. The graph is repeated in the Printed Answer Booklet.

Explain why the graph suggests that the given model is appropriate.

The owners believe that a = 120 and b = 1.15 are good estimates for the parameters in the model.

Show that the graph supports these estimates for the parameters.

Use the model S = 120 × 1.15^t to predict the number of items sold in the seventh month after

opening.

Use the model S = 120 × 1.15^t to predict the number of months after opening when the

total number of items sold after opening will first exceed 70 000.

Comment on how reliable this prediction may be.

An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is 5% less than the height of the step immediately below it.

In the painting the total height of the steps is 205 cm, correct to the nearest centimetre.

Determine the number of steps in the design.

In this question you must show detailed reasoning.

The cubic polynomial f(x) is defined by f(x) = 2x³ - 3x² - 11x + 6.

Use the factor theorem to show that (2 x - 1) is a factor of f (x).

Express f (x) in fully factorised form.

Hence solve the equation 2 × 8^y - 3 × 4^y - 11 × 2^y + 6 = 0.