Laws of logarithms

Show that the equation of the curve can be written in the form cos y = f (x), where f (x) is

fully simplified.

Use the substitution u = e^x - 2 to show that

The graph of y = 2^x can be transformed to the graph of y = 2^x+4 either by a translation or by a stretch.

Give full details of the translation.

Give full details of the stretch.

In this question you must show detailed reasoning.

Solve the equation log₂(8x) = 1 - log₂(1 - x).

Find the equation of this curve, giving your answer in the form e^3y = f(x).

Show that, when x = e², the y-coordinate of this curve can be written as

y = a + ln(be³ + c), where a, b and c are constants to be determined.

The diagram shows part of the curve y = ln (x - 4).

State the equation of the vertical asymptote to the curve y = ln (x - 4).

Find the total area enclosed by the curve y = ln (x - 4), the x-axis and the lines x = 4.5 x = 7. Give your answer in the form a ln 3 + b ln 2 + c where a, b and c are constants to be

found.

Find the exact area of the shaded region, giving your answer in the form p + q ln r, where p and q

are positive integers and r is a positive rational number.

In this question you must show detailed reasoning.

Find the x-coordinate of the point where the curve crosses the x axis.

The points A and B lie on the curve and have x coordinates 2 and 4. Show that the line AB is parallel to

the x-axis.

Find the coordinates of the turning point on the curve.

Determine whether this turning point is a maximum or a minimum.