For $y={\log }_{a}x$ to be defined, $a$ must be

Important Questions on Logarithm and its Applications

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $a={\log }_{2}3,b={\log }_{2}5$ and $c={\log }_{7}2$, then ${\log }_{140}63$ in terms of $a,b,c$ is

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The logarithm of $\frac{1}{256}$ to the base $2\sqrt{2}$ is

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $3^{x+1}=6^{{\log }_{2}3}$, then $x$ is:

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

${\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}$ is equal to

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $S=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{3}\right)-\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}\right)+\frac{1}{6}\left(\frac{1}{2^3}+\frac{1}{3^3}\right)\dots \dots \dots$ then $S=$

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $2^x{.3}^{x+4}=7^x$, then $x$ is equal to

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If  ${\log }_{\sqrt{3}} x+{\log }_{\sqrt[3]{5}} x+{\log }_{\sqrt[4]{5}} x+......$  up to $7$ terms $=35$, then the value of $x$ is :

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Find the value of ${\log }_5\sqrt{625}$

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If ${\log }_{2}6+\frac{1}{2x}={\log }_2\left(2^{\frac{1}{x}}+8\right)$ then the values of $x$ are

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $N=m!$ (where $m$ is a fixed positive integer $>2$), then $\frac{1}{{\log }_{2}N}+\frac{1}{{\log }_{3}N}+\frac{1}{{\log }_{4}N}+\mathrm{.....}+\frac{1}{{\log }_{m}N}=$

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If ${\log }_{4}x+{\log }_{16}x+{\log }_{64}x+{\log }_{256}x=\frac{25}{6}$ then the value of $x$ is

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

If $a^x=b^y=c^z=d^w$, the value of $x\left(\frac{1}{y}+\frac{1}{z}+\frac{1}{w}\right)$ is

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The solution of the equation $3^{{\log }_{a}x}+3x^{{\log }_{a}3}=2$ is given by

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The value of ${\log }_{2}32=$

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

The common difference of A.P. ${\log }_{2}2, {\log }_{2}4, {\log }_{2}8$ is

EASY

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

${\log }_{10}0.001=$ _____

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Let $f$ be a function defined on the set of all positive integers such that $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ for all positive integers $x, y.$ If $f\left(12\right)=24$ and $f\left(8\right)=15$ , the value of $f\left(48\right)$ is

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

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

HARD

Mathematics>Algebra>Logarithm and its Applications>Basics of Logarithm

Let $S$ be the sum of the digits of the number ${15}^2\times 5^{18}$ in base $10$. Then,