Embibe Experts Solutions for Chapter: Logarithm and Its Applications, Exercise 4: EXERCISE-4

Prove that $\frac{{\log }_{a}N}{{\log }_{ab}N}=1+{\log }_{a}b$ and indicate the permissible values of the letters:

Compute:  $a^{\frac{{\log }_b\left({\log }_{b}N\right)}{{\log }_{b}a}}$

Solve the system of equations : ${\log }_{a}x {\log }_a\left(xyz\right)=48$

                                                       ${\log }_{a}y{\log }_a\left(xyz\right)=12,a>0,a\ne 1$

                                                       ${\log }_{a}z{\log }_a\left(xyz\right)=84$

Solve for $x:$  $5^{\log x}+5x^{\log 5}=3(a>0);$ where base of log is $a$

Given: ${\log }_{10}34.56=1.5386$, find ${\log }_{10}3.456 ; {\log }_{10}0.3456 \text{and} {\log }_{10}0.003456$ -

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

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

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