Solve for $x:{\log }^2(4-x)+\log (4-x)·\log \left(x+\frac{1}{2}\right)-2{\log }^2\left(x+\frac{1}{2}\right)=0$.

Important Questions on Logarithm and Its Applications

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 25

Solve the following equation for $x$ and $y:{\log }_{100}|x+y|=\frac{1}{2}, {\log }_{10}y-{\log }_{10}\left|x\right|={\log }_{100}4$.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 26

Find all real numbers $x$ which satisfy the equation, $2{\log }_2{\log }_{2}x+{\log }_{1/2}{\log }_2\left(2\sqrt{2}x\right)=1$.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 27

Solve: ${\log }_{\frac{3}{4}}{\log }_8\left(x^2+7\right)+{\log }_{\frac{1}{2}}{\log }_{\frac{1}{4}}{\left(x^2+7\right)}^{-1}=-2$ 

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 28

Given : ${\log }_{10}34.56=1.5386$, find ${\log }_{10}3.456;{\log }_{10}0.3456\&{\log }_{10}0.003456$.

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 29

Find the number of positive integers which have the characteristic $3$, when the base of the logarithm is $7$.

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 30

If ${\log }_{10}2=0.3010\&{\log }_{10}3=0.4771$, find the value of ${\log }_{10}(2.25)$

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 31

Find the antilogarithm of $0.75$, if the base of the logarithm is $2401$.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-4>Q 32

Let       '$L$' denotes the antilog of $0.4$ to the base $1024$ .
and      '$M$' denotes the number of digits in $6^{10}$ (Given  ${\log }_{10}2=0.3010, {\log }_{10}3=0.4771$ )
and      '$N$' denotes the number of positive integers which have the characteristic $2$ , when base of the logarithm is $6$.
Find the value of  $LMN$.