If $\log_{(cos x)}(sin x) + \log_{(sin x)}(cos x) = 2,$ then the value of $x$ is

The set of real values of $x$ for which the inequality $\log _{27} 8 \leq \log _{3} x \lt 9^{\frac{1}{\log _{2} 3}}$ holds is

If $\log _{\left(x^{2}\right)} y+\log _{\left(y^{2}\right)} x=1$ and $y=x^{2}-30$, then the value of $x^{2}+y^{2}$ is ___________.

The approximate value of the expression $2\log_3 3n - \log_3(n^2 + 1)$ for a sufficiently large $n$ is

If $4^{\log_2{x}} - 4x + 9^{\log_3{y}} - 16y + 68 = 0$, then $y - x$ equals:

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

Let $a, b, c$ be real numbers greater than 1, and $n$ be a positive real number not equal to 1. If $log_n(log_2a) = 1; log_n(log_2b) = 2$ and $log_n(log_2c) = 3$ then which of the following is true?