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If $\log_{18} 24 = p$, then $\log_{96} 108$ equals

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

If $y=a+b \log _{e} x$ then which of the following is true?

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

If $\log_3(x^2 - 1)$, $\log_3(2x^2 + 1)$ and $\log_3(6x^2 + 3)$ are the first three terms of an arithmetic progression, then the sum of the next three terms of the progression is

The set of all values of $x$ satisfying the inequality $\log _{\left(x+\frac{1}{x}\right)}\left[\log _{2}\left(\frac{x-1}{x+2}\right)\right]>0$ is

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

Let $a = \dfrac{(\log_7 4)(\log_7 5 - \log_7 2)}{\log_{7} 25 (\log_7 8 - \log_7 4)}$. Then the value of $5^a$ is

If $\log_4 x = a$ and $\log_{25} x = b$, then $\log_x 10$ 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 product of the roots of the equation $\log_{2} 2^{(\log_{2}x)^{2}} -5 \log_{2}x+6=0$ 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?

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

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

If $\log_5(\log_8(x^2 - 1)) = 0$, then a possible value of $x$ 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

The inequality $\log_{2} \frac{3x - 1}{2 - x} < 1$ holds true for

If $x, y, z$ are positive real numbers such that $x^{12} = y^{16} = z^{24}$ and the three quantities $3 \log_y x, 4 \log_z y, n \log_x z$ are in arithmetic progression, then the value of $n$ is

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

The inequality $\log_{a}{f(x)} < \log_{a}{g(x)}$ implies that