BridgeEquations, logs and inequalitiesFree

Tangency and number of solutions of transcendental equations

mathematical

Consider the equation

\ln x = kx

, where

k

is a real constant.

The curve y = ln x with lines y = kx through the origin: the tangent line k = 1/e touching at (e, 1), a line with 0 < k < 1/e crossing twice, and a line with k < 0 crossing once.

Facts you may need

Tangent touches with equal value and equal gradient — $\ln a = ka$ and $\frac1a = k$.
$\ln x$ is increasing and concave — Shapes the number of crossings.

(a)

Find the value of

k

for which the line

y = kx

is a tangent to

y = \ln x

, and the point of contact.

(b)

State, with reasons, the number of solutions of

\ln x = kx

for all real

k