Logarithm and Antilogarithm Test 11

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Logarithm and Antilogarithm Test 11

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following real numbers is(are) non-positive?

A
$$log{ }_{ 0.3 }(\dfrac { \sqrt { 5 } +2 }{ \sqrt { 5 } -2 } )$$

B
$$log{ }_{ 7 }(\sqrt { 83 } -9\quad )$$

C
$$log{ }_{ 7\frac { \pi }{ 12 } }(cot\frac { \pi }{ 8 } \quad )$$

D
$${ log }_{ 2 }\sqrt { 9.\sqrt [ 3 ]{ { 27 }^{ \frac { -5 }{ 3 } }.243{ }^{ \frac { -7 }{ 5 } } } } $$

Solution

From given,

we have,

$$\log _{0.3}\left(\dfrac{\sqrt{5}+2}{\sqrt{5}-2}\right)$$

$$=\log _{0.3}\left(2+\sqrt{5}\right)-\log _{0.3}\left(\sqrt{5}-2\right)$$

$$=-2.39811$$
Question 2
1 / -0

$${ \left( \dfrac { 2 }{ 3 } \right) }^{ 0 }=?$$

A
$$\dfrac{3}{2}$$

B
$$\dfrac{2}{3}$$

C
$$1$$

D
$$0$$

Solution

Given $${ \left( \dfrac { 2 }{ 3 } \right) }^{ 0 }$$
By using tha law of exponents i.e., $$\left(\dfrac{a}{b}\right)^0=1$$
$$\therefore { \left( \dfrac { 2 }{ 3 } \right) }^{ 0 }=1$$
Question 3
1 / -0

If $$y$$ is any non-zero integer, then $$y^0$$ is equal to ....

A
$$1$$

B
$$0$$

C
$$-1$$

D
Not found

Solution

Using the law of exponents, $$a^0 =1$$

Therefore $$y^0=1$$
Question 4
1 / -0

The value of the logarithmic function $$\log_2 \log_2 \log_2 16$$ is equal to

A
$$0$$

B
$$1$$

C
$$2$$

D
$$4$$

Solution

Using, $$\log_a x^y = y\log_a x$$ and $$\log_a a = 1$$
$$\log _{ 2 }{ \log _{ 2 }{ \log _{ 2 }{ 16 } } } \\
=\log _{ 2 }{ \log _{ 2 }{ \log _{ 2 }{ { 2 }^{ 4 } } } } \\
=\log _{ 2 }{ \log _{ 2 }{ 4 } } \\
=\log _{ 2 }{ \log _{ 2 }{ { 2 }^{ 2 } } } \\
=\log _{ 2 }{ 2 } \\ =1\\
$$
Question 5
1 / -0

The value of $$\log_{15}225^{225}$$ is equal to

A
$$15$$

B
$$225$$

C
$$375$$

D
$$450$$

Solution

$$225\log _{15}{225}$$
$$=225\log _{15}{{15}^{2}}$$
$$=225\times 2\log _{15}{15} = 450$$
Question 6
1 / -0

If $$ \log_{10}4 = 0.6020$$, then the value of $$\log_{10}8$$ is equal to

A
$$0.803$$

B
$$0.853$$

C
$$0.903$$

D
$$0.953$$

Solution

$$\log_{10} 8 = \log_{10} (4 \times 2) = \log_{10} 4+ \log_{10} 2$$
$$=0.6020+0.3010=0.9030$$
Question 7
1 / -0

If $$\log_{10}(x+5) = 1$$, then value of $$x$$ is equal to

A
$$3$$

B
$$4$$

C
$$5$$

D
$$6$$

Solution

$$\log_{10}(x+5) = 1$$

$$ \mbox{In exponent form},$$

$$ (x+5) = { 10 }^{ 1 }$$

$$ \therefore x = 10 - 5 = 5 $$
Question 8
1 / -0

Given log 6 and log 8, then the only logarithm that cannot be obtained without using the table is

A
$$\log 64$$

B
$$\log 21$$

C
$$\log \frac{8}{3}$$

D
$$\log 9$$

Solution

Let $$\log { 6 } =x$$

$$\Rightarrow \log { 2 } +\log { 3 } =x$$ ....... $$(i)$$

Let $$\log { 8 } =y$$

$$\Rightarrow 3\log { 2 } =y$$
$$\Rightarrow \log 2=\dfrac{y}{3}$$ ...... $$(ii)$$
$$\therefore x=\log { 3 } +\dfrac { y }{ 3 } $$ ...... From $$(i)$$ and $$(ii)$$

$$\Rightarrow \log { 3 } =x-\dfrac { y }{ 3 } $$

A.
$$\log { 64 } =\log { { 2 }^{ 6 } }=6\log 2=6\dfrac{y}{3} $$

B.
$$\log { 21 } =\log { 3 } +\log {7} $$

Thus, $$\log 21$$ cannot be found from given values as we need the value of $$\log { 7 } $$.
C.
$$\log \dfrac{8}{3}=\log 8-\log 3=3\log 2-\log 3$$
D.
$$\log 9=\log 3^{2}=2\log 3$$
Hence, only $$\log 21$$ can't be obtained from given data.
Question 9
1 / -0

If $$\log (x + 1)+ \log (x - 1) = \log 3$$, then the value of $$x$$ will be

A
$$3$$

B
$$2$$

C
$$1$$

D
None of these

Solution

Given:
$$\log(x+1) +\log(x-1) = \log3$$
$$\implies \log (x+1)(x-1)=\log 3$$
Using multiplication rule of logarithms,
$$\log a+\log b = \log ab$$
Thus, $$\log(x^2-1) = \log3$$
$$\implies \Rightarrow x^2-1 = 3$$
$$\implies \Rightarrow x=\pm2$$
But $$x\neq-2 \quad \because x-1>0\Rightarrow x>1$$
$$\therefore x = 2$$
Hence, option C.
Question 10
1 / -0

The value of $$\log_{2\sqrt{3}}1728$$ is equal to

A
$$3$$

B
$$4$$

C
$$5$$

D
$$6$$

Solution

Let $$ \log _{ { 2\sqrt { 3 } } }{ 1728 } = x$$
$$\implies 1728 = \left( { 2\sqrt { 3 } } \right) { ^{ x } }\\ \implies (1728 = { 2 }^{ 6 }\times { 3 }^{ 3 } = { 2 }^{ 6 }{ \sqrt { 3 } }^{ 6 })\\ \therefore { 2 }^{ 6 }{ (\sqrt { 3 })}^{ 6 } = { (2\sqrt { 3 } ) }^{ x }\\ \therefore x = 6\\ \\ $$

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Logarithm and Antilogarithm Test 11

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Logarithm and Antilogarithm Test 11

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SHARING IS CARING

Question 1

$$log{ }_{ 0.3 }(\dfrac { \sqrt { 5 } +2 }{ \sqrt { 5 } -2 } )$$

$$log{ }_{ 7 }(\sqrt { 83 } -9\quad )$$

$$log{ }_{ 7\frac { \pi }{ 12 } }(cot\frac { \pi }{ 8 } \quad )$$

$${ log }_{ 2 }\sqrt { 9.\sqrt [ 3 ]{ { 27 }^{ \frac { -5 }{ 3 } }.243{ }^{ \frac { -7 }{ 5 } } } } $$

Solution

Question 2

$$\dfrac{3}{2}$$

$$\dfrac{2}{3}$$

$$1$$

$$0$$

Solution

Question 3

$$1$$

$$0$$

$$-1$$

Not found

Solution

Question 4

$$0$$

$$1$$

$$2$$

$$4$$

Solution

Question 5

$$15$$

$$225$$

$$375$$

$$450$$

Solution

Question 6

$$0.803$$

$$0.853$$

$$0.903$$

$$0.953$$

Solution

Question 7

$$3$$

$$4$$

$$5$$

$$6$$

Solution

Question 8

$$\log 64$$

$$\log 21$$

$$\log \frac{8}{3}$$

$$\log 9$$

Solution

Question 9

$$3$$

$$2$$

$$1$$

None of these

Solution

Question 10

$$3$$

$$4$$

$$5$$

$$6$$

Solution

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