Solve the equation 3e2x – 82ex + 27 = 0, giving your answers in the form k ln 3.

It is given that x = ln(2y – 3) – ln(y + 4). Express y in terms of x.

Use logarithms to solve the equation 3⁴ˣ⁺³ = 5²ˣ⁺⁷. Give your answer correct to 3 significant figures.

The positive numbers p and q are such that ln(p/q) = a and ln(q²p) = b. Express ln(p⁷q) in terms of a and b.

By first obtaining a quadratic equation in cos θ, solve the equation tan θ sin θ = 1 for 0° < θ < 360°.

The function f is defined by f(x) = −3x² + 2 for x ≤ −1.

At the point (4, −1) on a curve, the gradient of the curve is -3/2. It is given that dy/dx = x^(-1/2) + k, where k is…