Logarithmic Functions

$\frac{1}{{\log }_a b}\times \frac{1}{{\log }_{b }c}\times \frac{1}{{\log }_c a}$ is equal to __________.

If ${\log }_9\left(3{\log }_2\right(1+{\log }_3(1+2\log x)\left)\right)=\frac{1}{2}$ then find the value of $x.$

If ${\mathrm{x}}^3=1$ but $\mathrm{x}$ is not equal to $1$, then the value of $\log \left({\mathrm{x}}^{2015}+\frac{1}{{\mathrm{x}}^{2015}}+{\mathrm{x}}^{2016}+\frac{1}{{\mathrm{x}}^{2016}}\right)$ is

If ${\mathrm{x}}^3=1$ but $\mathrm{x}$ is not equal to $1$, then the value of $\log \left({\mathrm{x}}^{2015}+\frac{1}{{\mathrm{x}}^{2015}}+{\mathrm{x}}^{2016}+\frac{1}{{\mathrm{x}}^{2016}}\right)$ is

Two distinct positive integers a and b are such that, ${\mathrm{a}}^{\mathrm{b}}={\mathrm{b}}^{\mathrm{a}}$ and ${\mathrm{a}}^{\frac{\mathrm{b}}{\mathrm{a}}}=2$, then the value of $(\mathrm{a}+\mathrm{b})$ is:

If $\mathrm{h}\left(\mathrm{x}\right)=\frac{{\mathrm{e}}^{2\mathrm{x}}}{{\mathrm{e}}^{2\mathrm{x}}+\mathrm{e}}$, then the value of $\mathrm{h}\left(\frac{1}{4}\right)+\mathrm{h}\left(\frac{3}{4}\right)$ will be:

$\log (1+2+3)$ is exactly equal to

If $\log x+\log x^2 = 3$ , then the value of $x$is _____ .

Find the value of ${\log }_{3\sqrt{2}}324 :$

Find the value of :

$\frac{1}{{\log }_{0.5}\left(a\right)}+\frac{1}{{\log }_2\left(a\right)}$

Identify the natural logarithm among the following?

Find the value of ${\log }_4\left(2\right)-{\log }_4\left(\frac{1}{2}\right)$ _____.

Match the List $\mathrm{I}$ and List $\mathrm{II}$ and select the correct answer using the given code:

Find the value of ${\log }_4\sqrt{4}$?

If the $\log 1254=2.9275$, find the mantissa of $\log 12.54$ ______.

$Log\left({40}^2-{20}^2\right)$ is equal to-