If $\log_4 x = a$ and $\log_{25} x = b$, then $\log_x 10$ is

Suppose that a, b, and c are real numbers greater than 1. Then the value of $\dfrac{1}{1+\log_{a^2 b} \frac{c}{a}} + \dfrac{1}{1+\log_{b^2 c} \frac{a}{b}} + \dfrac{1}{1+\log_{c^2 a} \frac{b}{c}}$ 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 ___________.

Suppose that $\log_2[\log_3 (\log_4a)] = \log_3 [\log_4 (\log_2b)] = \log_4 [\log_2 (\log_3c)] = 0$ then the value of $a + b + c$ is

The value of $(\log_{3} 30)^{-1} + (\log_{4} 900)^{-1} + (\log_{5} 30)^{-1}$ 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

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