IPMAT Indore 2023 (SA) PYQs

If $\log_{25} [5 \log_3 (1+\log_3(1+2\log_2x))] = \frac12$ then $x$ is:

The numbers $2^{2024}$ and $5^{2024}$ are expanded and their digits are written out consecutively on one page. The total number of digits written on the page is

The value of $(0.04^{log_{\sqrt{5}}(\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...)})$ is __________.