Question 7 - A-Level Maths

Given that \(y = \log _ { a } x , x > 0\), where \(a\) is a positive constant,
1. (i) express \(x\) in terms of \(a\) and \(y\),
  (ii) deduce that \(\ln x = y \ln a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln a }\).
The curve \(C\) has equation \(y = \log _ { 10 } x , x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10 . Using the result in part (b),
find an equation for the tangent to \(C\) at \(A\). The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
Find the exact \(x\)-coordinate of \(B\).