Logarithm and Antilogarithm Test 8

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Logarithm and Antilogarithm Test 8

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Weekly Quiz Competition

Question 1
1 / -0

Simplify:
$$\left( { 4 }^{ -1 }\times { 3 }^{ -1 } \right) \div { 6 }^{ -1 }$$

A
$$\cfrac{1}{2}$$

B
$$\cfrac{1}{4}$$

C
$$\cfrac{2}{2}$$

D
None of these

Solution

we know,

$$a^m*b^m = (ab)^m$$
so,

$$4^{-1} * 3^{-1} = 12^{-1}$$

we know,

$$\dfrac{a^{-m}}{b^{-m}}=(\dfrac{a}{b})^{-m}=(\dfrac{b}{a})^{m}$$
so,
$$\dfrac{12^{-1}}{6^{-1}}=2^{-1}=\dfrac{1}{2}$$
Question 2
1 / -0

Simplify and give reasons:
$${(-2)}^{7}$$

A
$$-128$$

B
$$128$$

C
$$-28$$

D
None of these

Solution

$$(-2)^7$$

$$=(-1*2)^7$$

$$=(-1)^7(2)^7$$ $$\because (ab)^m=a^m*b^m$$

$$=-1*2^7$$

$$=-2^7$$

$$=-128$$
Question 3
1 / -0

Simplify the following:
$${ (-2) }^{ 7 }\times { (-2) }^{ 3 }\times { (-2) }^{ 4 }$$

A
$${(-2)}^{14}$$

B
$${(-2)}^{24}$$

C
$${(-2)}^{54}$$

D
None of these

Solution

we know,

$$a^{m}*a^{n}=a^{m+n}$$

$$\implies$$ $$a^{m}*a^{n}*a^{p}=a^{m+n+p}$$
so,

$${ (-2) }^{ 7 }\times { (-2) }^{ 3 }\times { (-2) }^{ 4 }$$

$$=(-2)^{7+3+4}$$

$$=(-2)^{14}$$
Question 4
1 / -0

Simplify and give reasons:
$$\cfrac { { 3 }^{ -2 } }{ 3 } \times \left( { 3 }^{ 0 }-{ 3 }^{ -1 } \right) $$

A
$$\cfrac{2}{81}$$

B
$$\cfrac{1}{81}$$

C
$$\cfrac{4}{81}$$

D
None of these

Solution
Question 5
1 / -0

Simplify the following:
$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 4 }\times { \left( \cfrac { 1 }{ 2 } \right) }^{ 5 }\times { \left( \cfrac { 1 }{ 2 } \right) }^{ 6 }$$

A
$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 15 }$$

B
$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 5 }$$

C
$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 10 }$$

D
None of these

Solution

we know,
$$a^{m}*a^{n}=(a)^{m+n}$$
so,
$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 4 }\times { \left( \cfrac { 1 }{ 2 } \right) }^{ 5 }$$
$$=\left(\cfrac12\right)^6$$
so,

$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 4+5+6 } $$
$$=\left(\cfrac12\right)^{15}$$
Question 6
1 / -0

Simplify and give reasons:
$$\left[ { \left( \cfrac { 3 }{ 2 } \right) }^{ -2 } \right] ^{ 2 }$$

A
$$\cfrac{16}{81}$$

B
$$\cfrac{6}{81}$$

C
$$\cfrac{16}{1}$$

D
None of these

Solution

we know that,

$$\because (a^m)^n=a^{mn}$$

so,

$$((\dfrac32)^{-2})^2=(\dfrac32)^{-4}$$
we also know

$$\because (\dfrac{a}{b})^{-m}=(\dfrac{b}{a})^m$$

$$=(\dfrac23)^4$$

$$=\dfrac{16}{81}$$
Question 7
1 / -0

Simplify and give reasons:
$${(-3)}^{-4}$$

A
$$\cfrac{1}{81}$$

B
$$\cfrac{-1}{81}$$

C
$$\cfrac{3}{81}$$

D
None of these

Solution

$$(-3)^{-4}$$

$$=(-1*3)^{-4}$$

$$=(-1)^{-4}(3)^{-4}$$ $$\because (ab)^m=a^m*b^m$$
$$=1*3^{-4}$$

$$=3^{-4}$$ $$\because a^{-m}=\dfrac1{a^m}$$

$$=(\dfrac13)^4$$

$$=\dfrac1{81}$$
Question 8
1 / -0

Simplify:
$$\left[ { \left( \cfrac { 3 }{ 5 } \right) }^{ -2 }\div { \left( \cfrac { 4 }{ 5 } \right) }^{ -3 } \right] \times { { \left( \cfrac { 3 }{ 5 } \right) } }^{ -2 }$$

A
$$\cfrac{{4}^{3}\times 5}{{3}^{4}}$$

B
$$\cfrac{{3}^{3}\times 5}{{3}^{4}}$$

C
$$\cfrac{{6}^{3}\times 5}{{3}^{4}}$$

D
None of these

Solution
Question 9
1 / -0

Exponential form of $$\log_{4}8 = x$$ is _____

A
$$x^{8} = 4$$

B
$$x^{4} = 8$$

C
$$4^{x} = 8$$

D
$$8^{x} = 4$$

Solution

We have, $$\log_{4}{8}=x$$

If $$ \log_{b}{a}=x$$ then $$a={b}^{x}$$

So, $${4}^{x}=8$$ is the correct answer.

Option $$C$$ is the correct answer.
Question 10
1 / -0

The logarithmic form of $${5}^{2}=25$$ is

A
$$\log _{ 5 }{ 2 } =25$$

B
$$\log _{ 2 }{ 5 } =25$$

C
$$\log _{ 5 }{ 25 } =2$$

D
$$\log _{ 25 }{ 5 } =2$$

Solution

$$5^2=25$$
Taking log with base $$5$$ both sides, we get
$$\log_55^2=\log_525$$
$$\Rightarrow \log_525=2\log_55$$
$$\Rightarrow \log_525=2$$ $$(\log_aa=1)$$
Hence, C is the correct option.

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Re-Attempt Weekly Quiz Competition

Logarithm and Antilogarithm Test 8

Result

Logarithm and Antilogarithm Test 8

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SHARING IS CARING

Question 1

$$\cfrac{1}{2}$$

$$\cfrac{1}{4}$$

$$\cfrac{2}{2}$$

None of these

Solution

Question 2

$$-128$$

$$128$$

$$-28$$

None of these

Solution

Question 3

$${(-2)}^{14}$$

$${(-2)}^{24}$$

$${(-2)}^{54}$$

None of these

Solution

Question 4

$$\cfrac{2}{81}$$

$$\cfrac{1}{81}$$

$$\cfrac{4}{81}$$

None of these

Solution

Question 5

$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 15 }$$

$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 5 }$$

$${ \left( \cfrac { 1 }{ 2 } \right) }^{ 10 }$$

None of these

Solution

Question 6

$$\cfrac{16}{81}$$

$$\cfrac{6}{81}$$

$$\cfrac{16}{1}$$

None of these

Solution

Question 7

$$\cfrac{1}{81}$$

$$\cfrac{-1}{81}$$

$$\cfrac{3}{81}$$

None of these

Solution

Question 8

$$\cfrac{{4}^{3}\times 5}{{3}^{4}}$$

$$\cfrac{{3}^{3}\times 5}{{3}^{4}}$$

$$\cfrac{{6}^{3}\times 5}{{3}^{4}}$$

None of these

Solution

Question 9

$$x^{8} = 4$$

$$x^{4} = 8$$

$$4^{x} = 8$$

$$8^{x} = 4$$

Solution

Question 10

$$\log _{ 5 }{ 2 } =25$$

$$\log _{ 2 }{ 5 } =25$$

$$\log _{ 5 }{ 25 } =2$$

$$\log _{ 25 }{ 5 } =2$$

Solution

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