Logarithm

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

Let $\alpha=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \infty$ and
$\beta=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots \infty$. Then the value of
$(0.2)^{\log _{\sqrt{5}}(\alpha)}+(0.04)^{\log _5(\beta)}$ is equal to :

The sum of all the real solutions of the equation $\log _{(x+3)}\left(6 x^2+28 x+30\right)=5-2 \log _{(6 x+10)}\left(x^2+6 x+9\right)$ is equal to :

The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{\mathrm{e}} x\right)^2+3}=x^8, x>0$, is :

If the solution of the equation \(\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right)\), is \(\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)\), where \(\alpha\), \(\beta\) are integers, then $$\alpha+\b...

Let \(S = \left\{ {\alpha :{{\log }_2}({9^{2\alpha - 4}} + 13) - {{\log }_2}\left( {{5 \over 2}.\,{3^{2\alpha - 4}} + 1} \right) = 2} \right\}\). Then the maximum value of \(\beta\) for which the equation $${x^2} - 2{\left( {\sum\limits_{...

The number of integral solutions \(x\) of \(\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^{2} \geq 0\) is :

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

The number of solutions of the equation log4(x \(-\) 1) = log2(x \(-\) 3) is _________.

The number of solutions of the equation \({\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0\), x > 0, is :

The number of distinct solutions of the equation
\({\log _{{1 \over 2}}}\left| {\sin x} \right| = 2 - {\log _{{1 \over 2}}}\left| {\cos x} \right|\) in the interval
[0, 2\(\pi\)], is ____.