Logarithmic Inequality

Q. If $x^2-4x+lo{g}_{\frac{1}{2}}a=0$ does not have two distinct real roots, then maximum value of a is

1. 
2. 
3. none of these
4. -

Q.

Solution set of the inequality
$\frac{1}{2^x-1}>\frac{1}{1-2^{x-1}}is$

1. $(0,lo{g}_2(4/3\left)\right)$

2. $(0,lo{g}_2(4/3\left)\right)\cup (1,\infty )$

3. $(1,\infty )$

4. $(-1,\infty )$

Q.

Find $x$ if ${\log }_2(x-5)>3$

1. x>5

2. x>13

3. x>8

4. x<13

Q.

Find $x$ if ${\log }_2(x-5)>3$

1. $x>15$

2. $x>13$

3. $x>18$

4. $x<13$

Q. If $x=lo{g}_5\left(1000\right)$ and $y=lo{g}_7\left(2058\right)$ then

1. x > y
2. x < y
3. x = y
4. None of these

Q. ${\log }_{4}2+{\log }_{4}4+{\log }_{4}x+{\log }_{4}16=6,$ then x =

1. 64
2. 4
3. 8
4. 32

Q. Given ${\log }_{10}x=a,{\log }_{10}y=b$

1. $\frac{10}{x}$
2. $10x$
3. $x$
4. $\frac{x}{10}$

Q. If $x+{\log }_{15}(1+3^x)=x{\log }_{15}5+{\log }_{15}12$, where $x$ is an integer, then what is $x$ equal to?

1. $-3$
2. $2$
3. $1$
4. $3$

Q. Prove that there do not exist natural numbers $x$ and $y,$ with $x>1,$ such that $\frac{x^7-1}{x-1}=y^5+1$.

Q. If $a^2=\log x,b^3=\log yand\frac{a^2}{2}-\frac{b^3}{3}=\log c$

Q. If $\log 2=x$ and $\log 3=y$, find out $\log \left(31104\right)$ in terms of $x$ and $y.$

Q. If $2\log x+3\log y=\log a$ and $x^2{y}^m=a$, then the value of $m$ is equal to

1. $3$
2. $1$
3. $5$
4. $6$

Q.

Find the value of ${27}^{\left(\frac{1}{{\log }_{4}3}\right)}+{16}^{{\log }_{4}2}+3^{\left(\frac{4}{{\log }_{7}9}\right)}$

1. 68

2. 117

3. 49

4. 64

Q. The number of solutions of $lo{g}_4(x-1)=lo{g}_2(x-3)$ is

1. 3
2. 1
3. 2
4. 0

Q. If $m=\log 20$ and $n=\log 25$, then the value of $x$, so that $2\log (x-4)=2m-n$ is equal to

1. $6$
2. $8$
3. $12$
4. $10$