Change of base or reciprocal relationship

Prove or use the relationship log_a(b) = 1/log_b(a) or convert between different logarithm bases.

4 questions · Moderate -0.1

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Edexcel AEA 2006 June Q3
11 marks Challenging +1.2
3．Given that \(x > y > 0\) ，
（a）by writing \(\log _ { y } x = z\) ，or otherwise，show that \(\log _ { y } x = \frac { 1 } { \log _ { x } y }\) ．
（b）Given also that \(\log _ { x } y = \log _ { y } x\) ，show that \(y = \frac { 1 } { x }\) ．
（c）Solve the simultaneous equations $$\begin{gathered} \log _ { x } y = \log _ { y } x \\ \log _ { x } ( x - y ) = \log _ { y } ( x + y ) \end{gathered}$$
OCR MEI Paper 2 2018 June Q5
3 marks Easy -1.2
5
  1. (A) Sketch the graph of \(y = 3 ^ { x }\).
    (B) Give the coordinates of any intercepts. The curve \(y = \mathrm { f } ( x )\) is the reflection of the curve \(y = 3 ^ { x }\) in the line \(y = x\).
  2. Find \(\mathrm { f } ( x )\).
Edexcel C3 Q7
10 marks Standard +0.3
  1. 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),
  2. find an equation for the tangent to \(C\) at \(A\). The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
  3. Find the exact \(x\)-coordinate of \(B\).
WJEC Unit 1 Specimen Q12
3 marks Moderate -0.5
12. Prove that $$\log _ { 7 } a \times \log _ { a } 19 = \log _ { 7 } 19$$ whatever the value of the positive constant \(a\).