Exponential and Logarithmic Functions

Let \(f(x) = \log_3{\frac{x}{2}} + \log_3{16} - \log_3{4}\), for \(x > 0\).

1. Show that \(f(x) = \log_3{2x}\).
2. Find the value of \(f(0.5)\) and of \(f(4.5)\).

The function \(f\) can also be written in the form \(f(x) = \frac{\ln{ax}}{\ln{b}}\)

3. 1. Write down the value of \(a\) and of \(b\).
   2. Hence on graph paper, sketch the graph of \(f\), for \(-5 \leq x \leq 5\), \(-5 \leq y \leq 5\), using a scale of \(1\) cm to \(1\) unit on each axis.
   3. Write down the equation of the asymptote.
4. Write down the valuw of \(f^{-1}(0)\).

The point \( \mathrm{A}\) lies on the graph of \(f\). At \(\mathrm{A}\), \(x =4.5\).

5. On your diagram, sketch the graph of \(f^{-1}\), noticing clearly the image of point \(\mathrm{A}\).