<p>Let&nbsp;\(f(x) = \log_3{\frac{x}{2}} + \log_3{16} - \log_3{4}\), for&nbsp;\(x &gt; 0\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(f(x) = \log_3{2x}\).</div></li><li><div class="list-item-paragraph">Find the value of \(f(0.5)\) and of \(f(4.5)\).</div></li></ol>
<p>The function \(f\) can also be written in the form \(f(x) = \frac{\ln{ax}}{\ln{b}}\)</p>
<ol class="abc-list" start="3"><li><ol class="i-list"><li><div class="list-item-paragraph">Write down the value of \(a\) and of \(b\).</div></li><li><div class="list-item-paragraph">Hence on graph paper, <strong>sketch</strong> the graph of \(f\), for&nbsp;\(-5 \leq x \leq 5\), \(-5 \leq y \leq 5\), using a scale of \(1\) cm to \(1\) unit on each axis.</div></li><li><div class="list-item-paragraph">Write down the equation of the asymptote.</div></li></ol></li><li><div class="list-item-paragraph">Write down the valuw of \(f^{-1}(0)\).</div></li></ol>
<p>The point \( \mathrm{A}\) lies on the graph of \(f\). At \(\mathrm{A}\), \(x =4.5\).</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">On your diagram, sketch the graph of \(f^{-1}\), noticing clearly the image of point \(\mathrm{A}\).</div></li></ol>

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

Find the value of \(f(0.5)\) and of \(f(4.5)\).

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

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

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

Exponential and Logarithmic Functions

International Baccalaureate Standard Level Mathematics