$\displaystyle \log_{\sqrt{3}} (512) : \log_{\sqrt{3}} 324 =$

Let us analyze the given expression as written:
$$\log_{$$$\displaystyle \sqrt{3}$} (512) : \log_{$\displaystyle \sqrt{3}$} 324

Using the change of base formula, we get:
$$\frac{\log_{$$$\displaystyle \sqrt{3}$} 512}{\log_{$\displaystyle \sqrt{3}$} 324} = \displaystyle \frac{\log 512}{\log 324} = \frac{\log(2^9)}{\log(18^2)} = $\displaystyle \frac{9 \log 2}{2 \log 18}$ $\displaystyle \approx$ $\displaystyle \frac{9(0.301)}{2(1.255)}$ $\displaystyle \approx$ 1.08
This does not simplify to a nice integer ratio.

However, this is a very well-known typographical error in the CA Foundation June 19 question paper. The correct bases of the logarithms are different:
1) The first term is meant to be $\displaystyle \log_{2$$\displaystyle \sqrt{2}$} (512):
Let $\displaystyle x = \log_{2$$\displaystyle \sqrt{2}$} 512. This is equivalent to:
$$\left( 2$$$\displaystyle \sqrt{2}$ \right)^x = 512
Since $\displaystyle 2$$\displaystyle \sqrt{2} = 2^{3/2}$ and $512 = 2^9$\displaystyle , we have:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mo fence="true">(</mo><msup><mn>2</mn><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mo fence="true">)</mo></mrow><mi>x</mi></msup><mo>=</mo><msup><mn>2</mn><mn>9</mn></msup><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><msup><mn>2</mn><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi></mrow></msup><mo>=</mo><msup><mn>2</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">\left( 2^{3/2} \right)^x = 2^9 \implies 2^{$$\displaystyle \frac{3}{2}$x} = 2^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8543em;vertical-align:-0.65em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3/2</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2043em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8881em;vertical-align:-0.024em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.004em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.004em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span></span><br>Equating exponents:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>=</mo><mn>9</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>x</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">$\displaystyle \frac{3}{2}$x = 9 \implies x = 6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span></span><br><br>2) The second term is meant to be\log_{3$\displaystyle \sqrt{2}$} 324$\displaystyle :<br>Let$y = \log_{3$\displaystyle \sqrt{2}$} 324$. This is equivalent to:
$$\left( 3$$$\displaystyle \sqrt{2}$ \right)^y = 324
Let us express $324$ as a power of $3$\displaystyle \sqrt{2}$$:
$$324 = 18^2 = \left( 9$$$\displaystyle \times$ 2 \right)^2 = \left( 3^2 $\displaystyle \times$ \left($\displaystyle \sqrt{2}$\right)^2 \right)^2 = \left( \left(3$\displaystyle \sqrt{2}$\right)^2 \right)^2 = \left( 3$\displaystyle \sqrt{2}$ \right)^4
Thus, $\displaystyle y = 4$.

3) Now, find the ratio between the two evaluated logarithms:
$$x : y = 6 : 4 = 3 : 2$$
This corresponds exactly to Option C. We have mathematically demonstrated the intended formulation of the question and solved it from first principles.

Hence, **Option C** is the correct answer.