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The sum of all possible real values of $x$ for which $\log_{x-3}(x^2-9)=\log_{x-3}(x+1)+2$, is

If $\log_{64} x^2 + \log_8 \sqrt{y} + 3\log_{512}\left(\sqrt{y}z\right) = 4$, where $x, y$ and $z$ are positive real numbers, then the minimum possible value of $(x+y+z)$ is:

The number of distinct integers $n$ for which $\log_{1/4}(n^2 - 7n + 11) > 0$, is

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \frac{4}{10^x} = \frac{81}{2}$, is

If a, b and c are positive real numbers such that $a > 10 \ge b \ge c$ and $\frac{\log_8(a+b)}{\log_2 c} + \frac{\log_{27}(a - b)}{\log_3 c} = \frac{2}{3}$, then the greatest possible integer value of a is

If $x$ is a positive real number such that $4 \log _{10} x+4 \log _{100} x+8 \log _{1000} x=13$, t hen greatest integer not exceeding x , is

For a real number $x$, if $\frac{1}{2}, \frac{\log _{3}\left(2^{x}-9\right)}{\log _{3} 4}$, and $\frac{\log _{5}\left(2^{x}+\frac{17}{2}\right)}{\log _{5} 4}$ are in an arithmetic progression, then the common difference is

For some positive real number $x$, if $\log_{\sqrt{3}}(x) + \frac{\log_x (25)}{\log_x (0.008)} = \frac{16}{3}$, then the value of $\log_3(3x^2)$ is

If $x$ and $y$ are positive real numbers such that $\log_{x}\left(x^{2}+12\right)=4$ and $3 \log _{y} x=1$, then $x+y$ equals

The number of distinct integer values of n satisfying $\frac{4-\log_2 n}{3-\log_4 n} < 0$, is

For a real number $a$, if $\frac{\log _{15} a+\log _{32} a}{\left(\log _{15} a\right)\left(\log _{32} a\right)}=4$ then a must lie in the range.

For all possible integers $n$ satisfying $2.25 \leq 2+2^{n+2} \leq 202$, the number of integer values of $3+3^{n+1}$ is

If $\log_2 [3 + \log_3 \{4 + \log_4 (x - 1)\}] - 2 = 0$, then $4x$ equals

If $5-\log _{10} \sqrt{1+x}+4 \log _{10} \sqrt{1-x}=\log _{10} \frac{1}{\sqrt{1-x^{2}}}$,then $100 \mathrm{x}$ equals

If $\log_a 30 = A$, $\log_a(5/3) = - B$ and $\log_2 a = 1 /3$, then $\log_3 a$ equals

$\small \dfrac{2\times4\times8\times16}{(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4}$ is equal to

The value of $\log_a \left(\frac{a}{b}\right) + \log_b \left(\frac{b}{a}\right)$, for $1 < a \leq b$ cannot be equal to

If y is a negative number such that $2^{y^2log_3 5} = 5^{\log_2 3}$, then y equals

If $\log _{4} 5=\left(\log _{4} y\right)\left(\log _{6} \sqrt{ } 5\right)$, then $y$ equals

The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

If $x$ is a real number, then $\sqrt{\log_e \frac{4x-x^2}{3}}$ is a real number if and only if

Let $x$ and $y$ be positive real numbers such that $\log _{5}(x+y)+\log _{5}(x-y)=3$, and $\log _{2} y-\log _{2} x=1-\log _{2} 3$. Then $x y$ equals

The smallest integer $n$ for which $4^n > 17^{19}$ holds, is closest to

If $p^3 = q^4 = r^5 = s^6$, then the value of $\log_s(pqr)$ is equal to

$\dfrac{1}{\log _{2} 100}-\dfrac{1}{\log _{4} 100}+\dfrac{1}{\log _{5} 100}-\dfrac{1}{\log _{10} 100}+\dfrac{1}{\log _{20} 100}-\dfrac{1}{\log _{25} 100}+\dfrac{1}{\log _{50} 100}= \ ?$

If $x$ is a positive quantity such that $2^x = 3^{\log_5 2}$, then $x$ is equal to

If $\log_{12}81 = p$, then $3\frac{4-p}{4+p}$ is equal to

If $\log _{2}\left(5+\log _{3} a\right)=3$ and $\log _{5}\left(4 a+12+\log _{2} b\right)=3$, then $a+b$ is equal to

If $x$ is a real number such that $\log_3 5 = \log_5 (2 + x)$, then which of the following is true?

If $\log (2^a \times 3^b \times 5^c)$ is the arithmetic mean of $\log (2^2 \times 3^3 \times 5)$, $\log (2^6 \times 3 \times 5^7)$, and $\log (2 \times 3^2 \times 5^4)$, then a equals