Embibe Experts Solutions for Chapter: Algebra, Exercise 6: EXERCISE - 1.6

The set of all real numbers $x$ for which ${\log }_{2004}\left({\log }_{2003}\left({\log }_{2002}\left({\log }_{2001}x\right)\right)\right)$ is defined is $\left\{\overline{)x}x>c\right\}$. If value of $c$ is in the form ${\left(200a\right)}^{200b}$, then the value of $ab$ is

There are $t$ distinct four-tuples $\left(a,b,c,d\right)$ of rational numbers with $a{\log }_{10}2+b{\log }_{10}3+c{\log }_{10}5+d{\log }_{10}7=2005$. What is $25\times t$?

Let $S$ be the set of ordered triples $\left(x,y,z\right)$ of real numbers for which ${\log }_{10}\left(x+y\right)=z$ and ${\log }_{10}\left(x^2+y^2\right)=z+1.$ There are real numbers $a$ and $b$ such that for all ordered triples $\left(x,y,z\right)$ in $S$, we have $x^3+y^3=a{10}^{3z}+b{10}^{2z}$. What is the value of $4\left(a+b\right)?$

Find the number of positive integers $x$ that satisfy the inequality $\left|3+{\log }_x\frac{1}{3}\right|<\frac{8}{3}$.

Suppose that $a, b$ and $c$ are real numbers greater than $1.$ Find the value of $\frac{1}{1+{\log }_{a^{2}b}\left({\displaystyle \frac{c}{a}}\right)}+\frac{1}{1+{\log }_{b^{2}c}\left({\displaystyle \frac{a}{b}}\right)}+\frac{1}{1+{\log }_{c^{2}a}\left({\displaystyle \frac{b}{c}}\right)}$

If $a>b>1$ and $\frac{1}{{\log }_{a}b}+\frac{1}{{\log }_{b}a}=\sqrt{1229}$ find the value of $\frac{1}{{\log }_{ab}b}-\frac{1}{{\log }_{ab}}.$

If $x, y>0, {\log }_{y}x+{\log }_{x}y=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2\sqrt{3}}$ is

If $b>1, x>0$ and ${\left(2x\right)}^{{\log }_{b}2}-{\left(3x\right)}^{{\log }_{b}3}=0$, then $\left(\frac{1}{x}\right)$ is