The value of $\{x \in R \mid \log_{10} ((1.6)^{1-x^2} - (0.625)^{6(1+x)}) \in R\}$ is

If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,then the number of values of $x$ which are integral multiples of $\frac{\pi}{4}$ is:

If $\log _{10} 7 = 0.8451$,then the position of the first significant figure of $7^{-20}$ is

Let $\log _a b=4$ and $\log _c d=2$,where $a, b, c, d$ are natural numbers. Given that $b-d=7$,the value of $c-a$ is:

Let $a, b, c$ be three distinct positive real numbers such that $(2a)^{\ln a} = (bc)^{\ln b}$ and $b^{\ln 2} = a^{\ln c}$. Then $6a + 5bc$ is equal to $........$.

The number of solutions of the equation $\frac{1}{2} \log _{\sqrt{3}}\left(\frac{x+1}{x+5}\right)+\log _{9}(x+5)^{2}=1$ is