Standard Logarithm

Q.

The number of elements in the set $x\in R:(\left|x\right|-3)\left|x+4\right|=6$

1. $2$

2. $1$

3. $3$

4. $4$

Q.

If $f\left(x\right)={\log }_{x^2}\left({\log }_{e}x\right)$, then $f\left(x\right)$ at $x=e$ is

1. $1$

2. $\frac{1}{e}$

3. $\frac{1}{2e}$

4. $0$

Q. If $y={\log }_{(7-a)}(2x^2+2x+a+3)$ is defined for all $x\in R,$ then possible integral value(s) of $a$ is (are)

1. $-3$
2. $-2$
3. $4$
4. $5$

Q. Integrate the function:
$\sqrt{1-4x^2}$

Q. The value of $3^{\sqrt{{\log }_{3}4}}$ is equal to

1. $3^{\sqrt{{\log }_{4}3}}$
2. $4^{\sqrt{{\log }_{4}3}}$
3. $4^{\sqrt{{\log }_{3}4}}$
4. $0$

Q. Find the value of the following expression: ${81}^{\frac{1}{{\log }_{5}3}}+{27}^{{\log }_{9}36}+3^{\frac{4}{{\log }_{7}9}}$

Q. Integrate the function: $\frac{1}{\sqrt{9-25x^2}}$

Q.

The product of two numbers is $-323.$ If one number is $17,$ then find the other number .

Q. If $a={\log }_{12}18$ & $b={\log }_{24}54$, then find the value of $ab+5(a-b)$.

Q. If $3^x=4^{x-1}$, then $x$ is equal to

1. $\frac{2{\log }_{3}2}{2{\log }_{3}2-1}$
2. $\frac{1}{1-{\log }_{4}3}$
3. $\frac{2}{2-{\log }_{2}3}$
4. $\frac{2{\log }_{2}3}{2{\log }_{2}3-1}$

Q. If ${\log }_3(2x^2+6x-5)>1$, then the number of integral values of $x$ which do not satisfy the inequality, is

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

Q. If P (A ∪ B) = 0.8 and P (A ∩ B) = 0.3, then P $\left(\overline{A}\right)$ + P $\left(\overline{B}\right)$ =
(a) 0.3
(b) 0.5
(c) 0.7
(d) 0.9

Q. Expand $\log \frac{343}{125}$

Q.

$\underset{x\to \infty }{lim}\frac{1^p+2^p+3^p+\dots \dots +n^p}{n^{p+1}}$ is

1. $\frac{1}{p}-\frac{1}{p-1}$

2. $\frac{1}{p+1}$

3. $\frac{1}{1-p}$

4. $\frac{1}{p+2}$

Q.

Find the value of $5^{{\log }_5{9}^2}$

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Q. Find the value of $3^{\frac{4}{{\log }_{2}9}}+{27}^{\frac{1}{{\log }_{49}9}}+{81}^{\frac{1}{{\log }_{4}3}}$

Q. If $a^2+4b^2=12ab$, then log(a + 2b) is

1. $log\frac{a}{2}+log\frac{b}{2}+log2$
2. $\frac{1}{2}[loga+logb-log2]$
3. $\frac{1}{2}[loga-logb+4log2]$
4. $\frac{1}{2}[loga+logb+4log2]$

Q. If $x^y=e^{x-y}$ then prove that $\frac{dy}{dx}=\frac{\log x}{(\log x+1{)}^2}$.

Q. If $x^y=e^{(x-y)}$, show that $\frac{dy}{dx}=\frac{\log x}{(1+\log x{)}^2}$.

Q. If $x={\log }_{2\sqrt{3}}1728$, then the value of x is

1. 5
2. 6
3. 7
4. 8

Q.

Find the value of ${\log }_{e}144$.

1. $2{\log }_{e}12$

2. $4{\log }_{e}2+2{\log }_{e}3$

3. $3{\log }_{e}2+{\log }_{e}18$

4. All of these

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. 64

2. 68

3. 49

4. 117

Q. If $a^2+b^2=7ab$, then prove that $\log \left(\frac{a+b}{3}\right)=\frac{1}{2}(\log a+\log b)$

Q. If ${\log }_{30}3=c$ and ${\log }_{30}5=d$, then the value of ${\log }_{30}8$ is

1. $3(1+c-d)$
2. $2(1-c-d)$
3. $3(1+c+d)$
4. $3(1-c-d)$

Q.

If $y=2^{\frac{1}{{\log }_{x}4}}$. Then

1. $y=x$

2. $y=x^2$

3. $y=x^{\frac{1}{2}}$

4. $y=x^3$

Q. $\int \log xdx=$

1. $x\left(\log x\right)+c$
2. $x(\log x-x)+c$
3. $x(\log x-1)+c$
4. $x(\log x+x)+c$

Q. Evaluate $\int \left[\right(\log x{)}^2-7\log x$ $+9]dx$

1. $x\left[\right(\log x{)}^2-9\log x]+c$
2. $x\left[\right(\log x{)}^2-9\left(\log x\right)+18]+c$
3. $x\left[\right(\log x{)}^2+9\log x-18]+c$
4. $x\left[\right(\log x{)}^2-7\log x+9]+c$

Q. Evaluate : ${\int }_0^1\log (\frac{1}{x}-1)dx$

Q. ${\int }_{-1/2}^{1/2}\cos x\log \left(\frac{1+x}{1-x}\right)dx=?$

Q. If $f\left(x\right)=\log \left\{\frac{u\left(x\right)}{v\left(x\right)}\right\},u\left(1\right)=v\left(1\right)\mathrm{and}u\text{'}\left(1\right)=v\text{'}\left(1\right)=2$, then find the value of f' (1).