Logarithms

If the domain of the function $f(x)=\sqrt{\log_{(0.6)}\left(\left|\dfrac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty)$, then the value of $a+b+c+d+e$ is ________.

Let $\alpha=\dfrac14+\dfrac18+\dfrac1{16}+\dots\infty$ and $\beta=\dfrac13+\dfrac19+\dfrac1{27}+\dots\infty$. Then the value of $(0.2)^{\log_{\sqrt5}(\alpha)}+(0.04)^{\log_5(\beta)}$ is equal to:

The sum of squares of all the real solutions of the equation $\log_{(x+1)}\left(2x^2+5x+3\right)=4-\log_{(2x+3)}\left(x^2+2x+1\right)$ is equal to ________.

Let $f$ be a polynomial function such that
$$\log_2(f(x))=\left(\log_2\left(2+\frac23+\frac29+\dots\infty\right)\right)\cdot\log_3\left(1+\frac{f(x)}{f(1/x)}\right),$$
$x>0$ and $f(6)=37$. Then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to ________.

Let $a = 3\sqrt{2}$ and $b = \dfrac{1}{5^{1/6}\sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that

$$3x + 2y = \log_a (18)^{\frac{5}{4}} \quad\text{and}$$
$$2x - y = \log_b\left(\sqrt{1080}\right),$$

then $4x + 5y$ is equal to _____.

Let $\alpha$ be a positive real number. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : (\alpha, \infty) \to \mathbb{R}$ be the functions defined by
$$f(x) = \sin\left(\frac{\pi x}{12}\right) \quad\text{and}\quad g(x) = \frac{2\log_e\left(\sqrt x - \sqrt\alpha\right)}{\log_e\left(e^{\sqrt x} - e^{\sqrt\alpha}\right)}.$$
Then the value of $\displaystyle\lim_{x\to\alpha^+} f(g(x))$ is _____.

The greatest integer less than or equal to
$$\int_1^2 \log_2(x^3+1)\,dx + \int_1^{\log_2 9} (2^x - 1)^{\frac13}\,dx$$
is _____.

The product of all positive real values of $x$ satisfying the equation
$$x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16}$$
is _____.

The domain of the function $f(x)=\dfrac{\cos^{-1}\left(\dfrac{x^2-5x+6}{x^2-9}\right)}{\log_e\left(x^2-3x+2\right)}$ is :

The sum of all the real roots of the equation $\left(e^{2x}-4\right)\left(6e^{2x}-5e^x+1\right)=0$ is

If the sum of all the roots of the equation $e^{2x}-11e^x-45e^{-x}+\dfrac{81}2=0$ is $\log_ep$, then $p$ is equal to ______.

Let $S_1=\left\{x\in\mathbf R-\{1,2\}:\dfrac{(x+2)\left(x^2+3x+5\right)}{-2+3x-x^2}\ge0\right\}$ and $S_2=\left\{x\in\mathbf R:3^{2x}-3^{x+1}-3^{x+2}+27\le0\right\}$. Then, $S_1\cup S_2$ is equal to :

Let $f(x)=\begin{cases}x^3-x^2+10x-7, & x\le1\\-2x+\log_2\left(b^2-4\right), & x>1\end{cases}$. Then the set of all values of b, for which $f(x)$ has maximum value at $x=1$, is :

The value of $\log_e2\,\dfrac{d}{dx}\left(\log_{\cos x}\operatorname{cosec}x\right)$ at $x=\dfrac\pi4$ is

The domain of the function $f(x)=\sin^{-1}\left[2x^2-3\right]+\log_2\left(\log_{\frac12}\left(x^2-5x+5\right)\right)$, where $[t]$ is the greatest integer function, is :

If $\alpha$, $\beta$ are the roots of the equation

$$x^2-\left(5+3^{\sqrt{\log_35}}-5^{\sqrt{\log_53}}\right)x+3\left(3^{(\log_35)^{\frac13}}-5^{(\log_53)^{\frac23}}-1\right)=0,$$

then the equation, whose roots are $\alpha+\dfrac1\beta$ and $\beta+\dfrac1\alpha$, is :

For |α| ≥ 1, let $5^{4-2\alpha}$, 63, $5^{2\alpha-1}$ be in A.P. with the common difference d. Let S$_n$ denote the sum of the first n terms of an A.P., whose first term is α$^2$ and the common difference is d. If S$_{30}$ − S$_{15}$ = 30k, then k is equal to

Let $m$ be the minimum possible value of $\log_3\left(3^{y_1} + 3^{y_2} + 3^{y_3}\right)$, where $y_1, y_2, y_3$ are real numbers for which $y_1 + y_2 + y_3 = 9$. Let $M$ be the maximum possible value of $(\log_3 x_1 + \log_3 x_2 + \log_3 x_3)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1 + x_2 + x_3 = 9$. Then the value of $\log_2(m^3) + \log_3(M^2)$ is ______

The value of

$$\left((\log_2 9)^2\right)^{\frac{1}{\log_2(\log_2 9)}} \times (\sqrt{7})^{\frac{1}{\log_4 7}}$$

is ______ .

If $3^x=4^{x-1}$, then $x=$