Logarithmic Function and its Properties

$x=\sqrt{a^{{\sin }^{-1}t}}, y=\sqrt{a^{{\cos }^{-1}t}}$,  what would be the value of $\frac{dy}{dx}$?

Solve the equation, ${\log }_3{\left(\frac{x-3}{x-7}\right)}^2+1={\log }_3\left(\frac{x-3}{x-1}\right)$

If ${\log }_{10}2=a \mathrm{and} {\log }_{2}3=b$, then express ${\log }_{4}100-2{\log }_{27}100$ in terms of $a$ and $b$.

A controlled experiment took place in a laboratory. Biologists estimated the number of microbes $N$ present in a culture $t$ days after the start of their experiment. The first results obtained are shown in the top two rows of the table below.

$\begin{array}{|lccccc|}\hline \mathit{t}& 3& 5& 10& 15& 20\\ \mathit{N}& 115& 178& 480& 900& 1600\\ \log N& & & & & \\ \hline\end{array}$

Complete the last row of the table, giving values to $2$ decimal places.

Draw a graph of $\log N$ against $t$, using appropriate  scales.

Find the domain of the function $f\left(x\right)={\log }_4(4x-3)$.

Find the domain of the function $f\left(x\right)={\log }_4(2x-3)$.

The value of $\left|\begin{array}{ll}{\log }_{5}729& {\log }_{3}5\\ {\log }_{5}27& {\log }_{9}25\end{array}\right|\times \left|\begin{array}{ll}{\log }_{3}5& {\log }_{27}5\\ {\log }_{5}9& {\log }_{5}9\end{array}\right|$ is

Find the solution set of the equation,${\text{log}}_{\frac{-{\text{x}}^2-6\text{x}}{10}}\left(\text{sin}3\text{x}+\text{sin x}\right)={\text{log}}_{\frac{-{\text{x}}^2-6\text{x}}{10}}\left(\text{sin}2\text{x}\right)$

If the solution set of the inequality $\frac{\left(\mathrm{l}\mathrm{o}{\mathrm{g}}_{3}x-2\right)\left(\mathrm{l}\mathrm{o}{\mathrm{g}}_{\frac{1}{2}}x-2\right)}{\left(\mathrm{l}\mathrm{o}{\mathrm{g}}_{4}x-1\right)}>0$ is $x\in \left(p, q\right)\cup \left(r, s\right),$ then the value of $p^2+(qr{)}^2. s$ is equal to

Find value of $x$ for which $\log \left(8-10\mathrm{x}-12{\mathrm{x}}^2\right)=3\log (2\mathrm{x}-1)$

Number of real solutions of the equation $x^{{\log }_{x}2}+x^2=3x$ is _____.

Which of the following is/are correct?

Let the product of all the solutions of the equation $\sqrt{\left({\log }_{3}2\sqrt{3x}+{\log }_x\sqrt[3]{3x}\right){\log }_3{x}^3}=2$ be $k$, then find the value of $18\mathrm{k}$

Suppose $n$ be an integer greater than $1$. Let $a_n=\frac{1}{{\log }_{n}2002}.$ Suppose $b=a_2+a_3+a_4+a_5$ and $c=a_{10}+a_{11}+a_{12}+a_{13}+a_{14}.$ Then find the value of $(c-b)$

The least integral value of $x$ satisfying the inequality ${\log }_{\frac{1}{5}}\left(x^2-6x+18\right)+2{\log }_5\left(x-4\right)<0$ is $\lambda$ then the value of  $3\lambda$ is

If the number of solutions of the equation ${\log }_{3}x {\log }_{4}x {\log }_{5}x={\log }_{3}x {\log }_{4}x+{\log }_{4}x {\log }_{5}x+{\log }_{5}x {\log }_{3}x$ is $\lambda$, then find the value of $16\lambda .$

Number of solution(s) of the equation $(x+1{)}^{{\log }_{x+1}\left(3+2x-x^2\right)}=(x-3\left)\right|x|$ is not equals to

If $9^{2{\log }_{9}x}+4x+3=0,$ then the number of real values of $x$ satisfying the equation is

If $\alpha$ is one of the solution of inequation ${\left({\log }_{10}100x\right)}^2+{\left({\log }_{10}10x\right)}^2+{\log }_{10}x\le 14$ then $\alpha$ can belongs to