Scientific Notations Of Real Numbers And Logarithms

The given number is 7 4 9 3 0 0 0 0 0 0 0 0 . (In integers decimal point at the end is usually omitted.)

Move the decimal point so that there is only one non - zero digit to its left.

The decimal point is to be moved 11 places to the left of its original position. So, the power of 10 is 11.

(The count of the number of digits between the old and new decimal point gives n the power of 10.)

Therefore, scientific notation is 7.49300000000×10¹¹ = 7.493×10¹¹.

The given number is 1 3 0 0 0 0 0 0 .

The decimal point is to be moved 7 places to the left of its original position. So the power of 10 is 7.

Therefore, scientific notation is 1.3000000×10⁷ = 1.3×10⁷

The given number is 1 0 5 0 0 3 .

The decimal point is to be moved 5 places to the left of its original position. So the power of 10 is 5.

Therefore,scientific notation is 1.05003×10⁵

The given number is 5 4 3 6 0 0 0 0 0 0 0 0 0 0 0 .

The decimal point is to be moved 14 places to the left of its original position. So the power of 10 is 14.

Therefore,scientific notation is 5.436×10¹⁴.

The given number is 0 . 0 0 9 6

The decimal point is to be moved 3 places to the right of its original position. So the power of 10 is - 3.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore,scientific notation is 9.6×10 ^{- 3}

The given number is 0 . 0 0 0 0 0 1 3 3 0 7

The decimal point is to be moved 6 places to the right of its original position. So the power of 10 is - 6.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 1.3307×10 ^{- 6}

The given number is 0 . 0 0 0 0 0 0 0 0 2 2

The decimal point is to be moved 9 places to the right of its original position. So the power of 10 is - 9.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 2.2×10 ^{- 9}

The given number is 0 . 0 0 0 0 0 0 0 0 0 0 0 0 9

The decimal point is to be moved 13 places to the right of its original position. So the power of 10 is - 13.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 9.0×10 ^{- 13}

The given number is 3.25 × 10^-6.

In this number the decimal number is 3.25

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is - 6.

So, the number in decimal form is 0.00000325

The given number is 4.134 × 10^-4

In this number the decimal number is 4.134

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is - 4.

So, the number in decimal form is 0.0004134

In decimal form, the given expression is written as:

4.134 × 10⁴

= 41.34 × 10³

= 413.4 × 10²

= 4134 × 10¹

= 41340

Hence, the decimal form of the given expression is: 41340

The given number is 1.86×10⁷.

In this number the decimal number is 1.86

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 7.

So, the number in becomes 18600000.00.

Therefore, the number in decimal form is 18600000.

The given number is 9.87×10⁹

In this number the decimal number is 9.87

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 9.

So, the number in becomes 9870000000.00

Therefore,the number in decimal form is 9870000000.

The given number is 1.432×10^-9

In this number the decimal number is 1.432

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 9.

So, the number indecimal form is 0.000000001432

In scientific notation,

1000 = (1.0×10³) and 20 = (2.0×10¹)⁶

∴(1000)²×(20)⁶ = (1.0×10³)²×(2.0×10¹)⁶

= (1.0)²×(10³)²×(2.0)⁶×(10¹)⁶

= 1×10⁶×64×10⁶

= 64×10¹²

= 6.4×10¹×10¹²

= 6.4×10¹³

∴ (1000)² x (20)⁶ in scientific notation is 6.4×10¹³

In scientific notation,

1500 = (1.5×10³) and 0.0001 = (1.0×10 ^{- 4})

∴(1500)³×(0.0001)² = (1.5×10³)³×(1.0×10 ^{- 4})²

= (1.5)³×(10³)³×(1.0)²×(10 ^{- 4})²

= 3.375 ×(10)⁹×1×(10) ^{- 8}

= 3.375×(10)¹

∴ (1500)³×(0.0001)² in scientific notation is 3.375×10¹

In scientific notation,

16000 = (1.6×10³) and 200 = (2.0×10²)

∴ (16000)³ (200)⁴ = (1.6×10⁴)³ ÷ (2.0×10²)⁴

∴ (16000)³ (200)⁴ in scientific notation is 2.56 ×103

In scientific notation,

0.003 = (3.0)×(10) ^{- 3}

0.0002 = (2.0)×(10) ^{- 4}

0.001 = (1.0)×(10) ^{- 3}

∴

⇒

= 6.9984×10 ^{- 28}

∴ (0.003)⁷(0.0002)⁵ ÷ (0.001)³ in scientific notation is 6.9984×10 - 28

(11000)³ (0.003)²( 30000)

Explanation: In scientific notation,

11000 = (1.1)×(10)⁴

0.003 = (3.0)×(10) ^{- 3}

30000 = (3.0)×(10)⁵

∴ (11000)³ (0.003)²( 30000)

⇒

1.331×10⁶×3×10 ^{- 5}

= 3.993×10¹

∴ (11000)³ (0.003)² ÷ (3000) in scientific notation is 3.993×10¹

(i) True

log₅125 = 3

⇒ 5³ = 125

(∵ x = log_ab is the logarithmic form of the exponential form a^x = b)

This is true.

(ii) False

⇒

(∵ x = log_ab is the logarithmic form of the exponential form ax = b)

Here

Therefore, this False.

(iii) False

Here its given log₄(6 + 3) = log₄6 + log₄3

Let us consider the RHS i.e.

log₄6 + log₄3 = log₄(6×3) (∵ according to the product rule loga(M×N) = logaM + logaN;

a,M,N are positive numbers,a≠1)

But here LHS is log4 (6 + 3)

Hence it’s False.

(iv) False

Here it’s given

Let us consider the LHS i.e.

(∵ log_aM ÷ log_aN = log_aM - log_aN

;a,M,N are positive numbers ,a≠1)

But here the RHS is

Hence both the sides are not equal and therefore it’s False.

(v) True

Here it’s given:

⇒ (∵ x = log_ab is the logarithmic form of the exponential form a^x = b)

⇒

Hence LHS = RHS

Therefore this is True.

(vi) False

Here it’s given that log_a (M - N) = log_a M ÷ log_aN

Let us consider the RHS

log_aM ÷ log_aN = log_aM - log_aN

(∵ according to quotient rule,log_aM ÷ log_aN = log_aM - log_aN ;a,M,N are positive numbers,a≠1)

But the LHS is log_a(M - N)

Therefore LHS≠RHS

Hence it’s False.

Here it’s given that 2⁴ = 16,

The given equation is in the form of a^x = b.

log_ab is the logarithmic form of the exponential form a^x = b

In the equation 2⁴ = 16 (a = 2,b = 16 ,x = 4)

⇒ log₂16 = 4

Here it’s given that 3⁵ = 243

The given equation is in the form of a^x = b.

log_ab is the logarithmic form of the exponential form a^x = b

In the equation 3⁵ = 243 (a = 3,b = 343 ,x = 5)

⇒ log₃243 = 5

Here it’s given that 10 ^{- 1} = 0.1

The given equation is in the form of a^x = b.

log_ab is the logarithmic form of the exponential form a^x = b

In the equation 10 ^{- 1} = 0.1 (a = 10,b = 0.1,x = - 1)

⇒

Here it’s given that

The given equation is in the form of a^x = b.

log_ab is the logarithmic form of the exponential form a^x = b

In the given equation (a = 8,,)

⇒

Here it’s given that

The given equation is in the form of a^x = b.

log_ab is the logarithmic form of the exponential form a^x = b

In the given equation (a = 25,b = 5,)

⇒

Here it’s given that

The given equation is in the form of a^x = b.

log_ab is the logarithmic form of the exponential form a^x = b

In the equation (a = 12,,x = - 2)

⇒

Here it’s given that log₆216 = 3

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation log₆216 = 3 ( a = 6,b = 216 ,x = 3)

⇒ 6³ = 216

Here it’s given that

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation ( a = 9,b = 3 ,)

⇒

Here it’s given that log₅1 = 0

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation log₅1 = 0 (a = 5,b = 1,x = 0)

⇒ 5⁰ = 1

Here it’s given that

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation (,b = 9,x = 4)

⇒

Here it’s given that

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation ( a = 64,,)

⇒

Here it’s given that log_0.58 = - 3

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation log_0.58 = - 3 (a = 0.5,b = 8,x = - 3)

⇒ (0.5) ^{- 3} = 8

i.e. log₃(3 ^{- 4}) = - 4(log₃3)

(∵ nlog_aM = log_aMⁿ)

⇒ - 4(1) = - 4

(log_aa = 1)

log₇343 = log₇7³

⇒ 3log₇7 (∵ nlog_aM = log_aMⁿ)

⇒ 1(∵ log_aa = 1)

log₆6⁵

⇒ 5log₆6

(∵ nlog_aM = log_aMⁿ)

= 5(1)

(∵ log_aa = 1)

= 5

Here we have i.e.

⇒ , here is

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒ 3( - 1) = - 3

Here we have log100.0001, i.e.

⇒

⇒ - 4log₁₀10 (∵ nlog_aM = log_aMⁿ)

⇒ - 4(1) = - 4 (∵ log_aa = 1)

Here we have ,

⇒

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒

⇒

⇒ x = 5

Hence the value of is 5.

⇒ i.e

⇒

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

Or

Or

log₃y = - 2

log₃y = - 2

⇒ 3 ^{- 2} = y

⇒ y = 3 ^{- 2}

⇒

i.e.

⇒

⇒

⇒

∴

log_x0.001 = - 3

⇒ x ^{- 3} = 0.001

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒

⇒

⇒ x = 10

x + 2log₂₇9 = 0

⇒ x = - 2log₂₇9

⇒ x = log₂₇9 ^{- 2}

⇒ x = log₃³(3²) ^{- 2}

⇒ x = log₃³(3) ^{- 4}

⇒ (3³)^x = 3 ^{– 4}

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒ 3x = - 4 (compare the exponents)

⇒

log₁₀3 + log₁₀3 = log₁₀(3×3) = log₁₀9

(∵ using the product rule,log_a(M×N) = (log_aM) + (log_aN);a,M,N are positive numbers ,a≠1)

(using the quotient rule log_a(M ÷ N) = (log_aM) - (log_aN) );a,M,N are positive numbers ,a≠1)

=

log₇21 + log₇77 + log₇88 - log₇121 - log₇24

⇒

(using the product rule and the quotient rule i.e.

log_a(M×N) = (log_aM) + (log_aN) and

log_a(M ÷ N) = (log_aM) - (log_aN))

⇒

⇒

⇒ log₇7²

⇒ 2log₇7 = 2

(∵ log₇7 = 1 )

⇒ log₈(16×52) - log₈13

(∵ log_a(M×N) = (log_aM) + (log_aN) and )

⇒

(∵ loga(M ÷ N) = (log_aM) - (log_aN))

⇒ log₈(16×4) = log₈64

⇒ 8x = 64 or x = 2

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

5log₁₀2 + 2log₁₀3 - 6log₆₄4

Here log₆₄4 = x

⇒ 64^x = 4

⇒ (4³)^x = 4

⇒ 3x = 1

⇒

∴

∴ 5log₁₀2 + 2log₁₀3 - 6log₆₄4 = log₁₀2⁵ + log₁₀3² - 2

= log₁₀32 + log₁₀9 - 2log₁₀10

= log₁₀32 + log₁₀9 - log₁₀10²

⁼

log₁₀8 + log₁₀5 - log₁₀4

⇒ log₁₀(8×5) - log₁₀4

(∵ log_a(M×N) = (log_aM) + (log_aN))

⇒

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

⇒ log10(2×5) = log1010 = 1

(∵ log_aa = 1)

log₄(x + 4) + log₄8 = 2

⇒ log₄((x + 4)×8) = 2

⇒ log₄(8x + 32) = 2

⇒ 8x + 32 = 4²

⇒ 8x + 32 = 16

⇒ 8x = 16 - 32 = - 16

⇒ 8x = - 16

⇒ x = - 2

log₆(x + 4) - log₆(x - 1) = 2

⇒

⇒ (x + 4)(x - 1) = 6² = 6×6

⇒ x + 4 = 6

⇒ x = 6 - 4 = 2

log₂x + log₄x + log₈x =

Here LHS is

⇒ log₂x + log₂²x + log₂³x

⇒

(∵)

⇒

(∵ log_aMⁿ = nlog_aM)

⇒

⇒

⇒

Now we equate LHS to the RHS i.e.

⇒

⇒ logx2 = 1 or x1 = 2 or x = 2

log₄(8log₂x) = 2

⇒ 8log₂x = 4²

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒ log₂x⁸ = 16

(∵ log_aMⁿ = nlog_aM)

⇒ 2¹⁶ = x⁸

⇒ (2²)⁸ = x⁸

⇒ x = 2² = 4

log₁₀5 + log₁₀(5x + 1) = log₁₀(x + 5) + 1

⇒ log₁₀(5(5x + 1)) - log₁₀(x + 5) = 1

⇒

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

⇒

⇒ 25x + 5 = 10(x + 5)

⇒ 25x + 5 = 10x + 50

⇒ 25x - 10x = 50 - 5 = 45

⇒ 15x = 45

⇒ x = 3

4log₂x - log₂5 = log₂125

⇒ log₂x⁴ - log₂5 = log₂125

⇒

⇒

⇒ x⁴ = 5×125 = 5×5³ = 5⁴

⇒ x = 5

log₃25 + log₃x = 3log₃5

⇒ log₃(25×x) = 3log₃5

⇒ log₃(25x) = log₃5³

⇒ 25x = 5³ or (5²)x = 5³

⇒ x = 5

⇒

⇒

(∵ log_a(M ÷ N) = log_aM - log_aN)

⇒

(∵ a^x = b is the exponential form of logarithmic form log_ab)

⇒

⇒

⇒ 5x - 2 = 3(x + 4)

⇒ 5x - 2 = 3x + 12

⇒ 5x - 3x = 12 + 2

⇒ 2x = 14

⇒ x = 7

(i) log_a15 = log_a(5×3)

i.e. log_a(5×3) = log_a5 + log_a3

(∵ log_a(M×N) = (log_aM) + (log_aN))

= z + y(∵ log_a5 = z,log_a3 = y)

(ii) log_a8 = log_a2³ = 3log_a2 = 3x

(∵ log_a2 = x)

(iii) log_a30 = log_a(5×3×2) = log_a(5) + log_a(3) + log_a(2)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= z + y + x

(∵ log_a5 = z,log_a3 = y,log_a2 = x)

= x + y + z

(iv)

⇒ log_a(3×3×3) - log_a(5×5×5)

⇒ (log_a3 + log_a3 + log_a3) - (log_a5 + log_a5 + log_a5)

⇒ (y + y + y) - (z + z + z) = 3y - 3z = 3(y - z)

(v)

⇒ log_a10 - log_a3

(∵ log_a(M ÷ N) = log_aM - log_aN)

Here log_a10 = log_a(5×2)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= log_a5 + log_a2 = z + x (∵ log_a5 = z,log_a2 = x)

(vi)

⇒

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= y - x(∵ log_a3 = y,log_a2 = x)

log₁₀1600 = 2 + 4log₁₀2 = 2log₁₀10 + 4log₁₀2

Let us consider the RHS:

i.e. 2 + 4log₁₀2 = 2log₁₀10 + 4log₁₀2

(∵ log_aa = 1)

= log₁₀10² + log₁₀2⁴

(∵ log_aMⁿ = nlog_aM)

= log₁₀100 + log₁₀16

= log₁₀(100×16)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= log₁₀1600

Hence LHS = RHS

log₁₀12500 = 2 + 3log₁₀5 = 2log₁₀10 + 3log₁₀5

Let us consider the RHS:

i.e. 2 + 3log₁₀5 = 2log₁₀10 + 3log₁₀5

= log₁₀10² + log₁₀5³

(∵ log_aMⁿ = nlog_aM)

= log₁₀(10²×5³)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= log₁₀(100×125)

= log₁₀(12500)

Hence LHS = RHS

log₁₀2500 = 4 - 2log₁₀2

Let us consider the RHS:

i.e. 4 - 2log₁₀2 = 4log₁₀10 - 2log₁₀2

= log₁₀10⁴ - log₁₀2²

(∵ log_aMⁿ = nlog_aM)

=

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

Hence LHS = RHS

log₁₀0.16 = 2log₁₀4 - 2

Let us consider the RHS:

i.e. 2log₁₀4 - 2 = 2log₁₀4 - 2log₁₀10

= log₁₀4² - log₁₀10²

(∵ log_aMⁿ = nlog_aM)

=

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= log₁₀(0.16) = log₁₀0.16

Hence LHS = RHS

log₅0.00125 = 3 - 5log₅10

Let us consider the RHS:

i.e. 3 - 5log₅10 = 3log₅5 - 5log₅10(∵ log_aa = 1)

= log₅5³ - log₅10⁵

(∵ log_aMⁿ = nlog_aM)

=

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= log₅0.00125

Let us consider the RHS

(∵)

₌ log₅6 - log₅2 + 4

= log₅6 - log₅2 + 4log₅5

(∵ log_a(M ÷ N) = ( log_aM) - (log_aN) and log_a(M×N) = (log_aM ) + (log_aN))

= log₅1875

Hence LHS = RHS

Scientific Notation: A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

Let N = 92.43

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 92.43 = 9.243 × 10¹

Scientific Notation: A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

Let N = 0.9243

Divide N by 10000 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.9243 = 9.243 × 10^–1

Scientific Notation: A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

Let N = 9243

Multiply and Divide N by 1000, we get

Thus, scientific notation of 9243 = 9.243 × 10³

Scientific Notation: A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

Let N = 924300

Multiply and Divide N by 10⁵, we get

Thus, scientific notation of 924300 = 9.243 × 10⁵

Let N = 0.009243

Divide N by 10⁶ to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.009243 = 9.243 × 10^–3

Scientific Notation: A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

Let N = 0.09243

Divide N by 10⁵ to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.09243 = 9.243 × 10^–2

Characteristic: In a scientific number, the power of 10 determines the characteristic.

Let N = 4576

Multiply and Divide N by 1000, we get

Thus, scientific notation of 4576 = 4.576 × 10³

Consider,

log 4576 = log (4.576 × 10³ )

= log 4.576 + log 10³

(since, log (a×b) = log a + log b)

= log 4.576 + 3 (since, log 10ⁿ = n)

Thus characteristic of log 4576 is 3

Characteristic: In a scientific number, the power of 10 determines the characteristic.

Let N = 24.56

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 24.56 = 2.456 × 10¹

Consider,

log 24.56 = log (2.456 × 10¹ )

= log 2.456 + log 10¹

(since, log (a×b) = log a + log b)

= log 2.456 + 1 (since, log 10ⁿ = n)

Thus characteristic of log 24.56 is 1

Characteristic: In a scientific number, the power of 10 determines the characteristic.

Let N = 0.00257

Divide N by 10⁵ to remove decimal, we get

Multiply and Divide N by 100, we get

Thus, scientific notation 0.00257 = 2.57 × 10^–3

Consider,

log 0.00257 = log (2.57 × 10^–3 )

= log 2.57 + log 10^–3

(since, log (a×b) = log a + log b)

= log 2.57 + (–3)

(since, log 10ⁿ = n)

Thus characteristic of log 0.00257 is –3

Characteristic: In a scientific number, the power of 10 determines the characteristic.

Let N = 0.0756

Divide N by 10⁴ to remove decimal, we get

Multiply and Divide N by 100, we get

Thus, scientific notation 0.0756 = 7.56 × 10^–2

Consider,

log 0.0756 = log (7.56 × 10^–2 )

= log 7.56 + log 10^–2

(since, log (a×b) = log a + log b)

= log 7.56 + (–2)

(since, log 10ⁿ = n)

Thus characteristic of log 0.0756 is –2

Characteristic: In a scientific number, the power of 10 determines the characteristic.

Let N = 0.2798

Divide N by 10⁴ to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.2798 = 2.798 × 10^–1

Consider,

log 0.2798 = log (2.798 × 10^–1 )

= log 2.798 + log 10^–1

(since, log (a×b) = log a + log b)

= log 2.798 + (–1)

(since, log 10ⁿ = n)

Thus characteristic of log 0.2798 is –1

Characteristic: In a scientific number, the power of 10 determines the characteristic.

Consider,

log 6.453 = lo

g (6.453 × 10⁰ )

= log 6.453 + log 10⁰

(since, log (a×b) = log a + log b)

= log 6.453 + 0

(since, log 10ⁿ = n)

Thus characteristic of log 6.453 is 0

Mantissa: Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23750

Multiply and Divide N by 10000, we get

Thus, scientific notation of 23750 = 2.3750 × 10⁴

Consider,

log 23750 = log (2.3750 × 10⁴ )

= log 2.375 + log 10⁴

(since, log (a×b) = log a + log b)

= log 2.375 + 4

(since, log 10ⁿ = n)

Thus characteristic of log 23750 is 4

Thus, Value of log 23750 = 4 + 0.3756 = 4.3756

Mantissa: Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23.75

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 23.75 = 2.375 × 10¹

Consider,

log 23.75 = log (2.375 × 10¹ )

= log 2.375 + log 10¹

(since, log (a×b) = log a + log b)

= log 2.375 + 1

(since, log 10ⁿ = n)

Thus characteristic of log 23.75 is 1

Thus, Value of log 23.75 = 1 + 0.3756 = 1.3756

Mantissa: Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Consider,

log 2.375 = log (2.375 × 10⁰ )

= log 2.375 + log 10⁰

(since, log (a×b) = log a + log b)

= log 2.375 + 0

(since, log 10ⁿ = n)

Thus characteristic of log 2.375 is 0

Thus, Value of log 2.375 = 0 + 0.3756 = 0.3756

Mantissa: Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 0.2375

Divide N by 10000 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.2375 = 2.375 × 10^–1

Consider,

log 0.2375 = log (2.375 × 10^–1 )

= log 2.375 + log 10^–1

(since, log (a×b) = log a + log b)

= log 2.375 + (–1)

(since, log 10ⁿ = n)

Thus characteristic of log 0.2375 is –1

Thus, Value of log 0.2375 = –1 + 0.3756 = ̅1.3756

Mantissa: Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23750000

Multiply and Divide N by 10⁷, we get

Thus, scientific notation 23750000 = 2.375 × 10⁷

Consider,

log 23750000 = log (2.375 × 10⁷ )

= log 2.375 + log 10⁷

(since, log (a×b) = log a + log b)

= log 2.375 + 7

(since, log 10ⁿ = n)

Thus characteristic of log 23750000 is 7

Thus, Value of log 23750000 = 7 + 0.3756 = 7.3756

Mantissa: Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 0.00002375

Divide N by 10⁸ to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.00002375 = 2.375 × 10^–5

Consider,

log 0.00002375 = log (2.375 × 10^–5 )

= log 2.375 + log 10^–5

(since, log (a×b) = log a + log b)

= log 2.375 + (–5)

(since, log 10ⁿ = n)

Thus characteristic of log 0.00002375 is –5

Thus, Value of log 0.00002375 = –5 + 0.3756 = ̅5.3756

Let N = 23.17

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 23.17 = 2.317 × 10¹

Consider,

log 23.17 = log (2.317 × 10¹ )

= log 2.317 + log 10¹

(since, log (a×b) = log a + log b)

= log 2.317 + 1

(since, log 10ⁿ = n)

Thus characteristic of log 23.17 is 1

From the table log 2.31 = 0.3636

Mean difference of 7 is 0.0013

Thus, Mantissa of log 23.17 = 0.3636 + 0.0013 = 0.3649

Thus, Value of log 23.17 = 1 + 0.3649 = 1.3649

Let N = 9.321

Consider,

log 9.321 = log (9.321 × 10⁰ )

= log 9.321 + log 10⁰

(since, log (a×b) = log a + log b)

= log 9.321 + 0

(since, log 10ⁿ = n)

Thus characteristic of log 9.321 is 0

From the table log 9.32 = 0.9694

Mean difference of 1 is 0

Thus, Mantissa of log 9.321 = 0.9694

Thus, Value of log 9.32 = 0+ 0.9694 = 0.9694

Let N = 329.5

Divide N by 10 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 329.5 = 3.295 × 10²

Consider,

log 329.5 = log (3.295 × 10² )

= log 3.295 + log 10²

(since, log (a×b) = log a + log b)

= log 3.295 + 2

(since, log 10ⁿ = n)

Thus characteristic of log 329.5 is 2

From the table log 3.29 = 0.5172

Mean difference of 5 is 0.0007

Thus, Mantissa of log 329.5 = 0.5172+0.0007 = 0.5179

Thus, Value of log 329.5 = 2+0.5178 = 2.5179

Let N = 0.001364

Divide N by 10⁶ to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.001364 = 1.364 × 10^–3

Consider,

log 0.001364 = log (1.364 × 10^–3 )

= log 1.364 + log 10^–3

(since, log (a×b) = log a + log b)

= log 1.364 + (–3)

(since, log 10ⁿ = n)

Thus characteristic of log 1.364 is –3

From the table log 1.36 = 0.1335

Mean difference of 4 is 0.0013

Thus, Mantissa of log 0.001364 = 0.1335+0.0013 = 0.1348

Thus, Value of log 0.001364 = –3 + 0.1348 = ̅3.1348

Let N = 0.9876

Divide N by 10⁴ to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.9876= 9.876 × 10^–1

Consider,

log 0.9876 = log (9.876 × 10^–1 )

= log 9.876 + log 10^–1

(since, log (a×b) = log a + log b)

= log 9.876 + (–1)

(since, log 10ⁿ = n)

Thus characteristic of log 0.9876 is –1

From the table log 9.87=0.9943

Mean difference of 6 is 0.0003

Thus, Mantissa of log 0.9876 = 0.9943+0.0003=0.9946

Thus, Value of log 0.9876 = –1+0.9946 = ̅1.9946

Let N = 6576

Multiply and Divide N by 1000, we get

Thus, scientific notation 6576= 6.576 × 10³

Consider,

log 6576 = log (6.576 × 10³ )

= log 6.576 + log 10³

(since, log (a×b) = log a + log b)

= log 6.576 + 3

(since, log 10ⁿ = n)

Thus characteristic of log 6576 is 3

From the table log 6.57=0.8176

Mean difference of 6 is 0.0004

Thus, Mantissa of log 6576 = 0.8176 +0.0004=0.8180

Thus, Value of log 6576 = 3+0.8180 = 3.8180

(i) Characteristic is 3

Mantissa is 0.072

From the antilog table antilog 0.072 = 1.180

Now as the characteristic is 3, therefore we will place the decimal after 3+1=4 numbers in 1180

∴ antilog 3.072 = 1180

(ii) Characteristic is 1

Mantissa is 0.759

From the antilog table antilog 0.759 = 5.741

Now as the characteristic is 1, therefore we will place the decimal after 1+1=2 numbers in 5741

∴ antilog 1.759 = 57.41

(iii) Characteristic is ̅1 = –1

Mantissa is 0.3826

From the antilog table antilog 0.382 = 2.410

Mean Value of 6 is 0.003

Thus, antilog 0.3826 = 2.410+0.003 = 2.413

Now as the characteristic is –1, therefore we will move decimal

–1+1=0 places left in 2.413

∴ antilog ̅1.3826 = 0.2413

(iv) Characteristic is ̅3 = –3

Mantissa is 0.6037

From the antilog table antilog 0.603 = 4.009

Mean Value of 7 is 0.006

Thus, antilog 0.6037 = 4.009+0.006 = 4.015

Now as the characteristic is –3,

therefore we will move decimal

–3+1=2 places left in 4.015

∴ antilog ̅3.6037 = 0.004015

(v) Characteristic is 0

Mantissa is 0.2732

From the antilog table antilog 0.273 = 1.875

Mean value 2 is 0.001

Thus, antilog 0.2732 = 1.875+0.001 = 1.876

Now as the characteristic is 0, therefore we will place the decimal after 0+1=1 numbers in 1876

∴ antilog 0.2732 = 1.876

(vi) Characteristic is ̅2 = –2

Mantissa is 0.1798

From the antilog table antilog 0.179 = 1.510

Mean Value of 8 is 0.003

Thus, antilog 0.1798 = 1.510+0.003 = 1.513

Now as the characteristic is –2, therefore we will move decimal

–2+1=1 places left in 1.513

∴ antilog ̅2.1798 = 0.01513

Let x = 816.3 × 37.42

Taking log on both side we get,

⇒ logx = log (816.3 × 37.42)

= log 816.3 + log 37.42 (since, log a× b = log a + log b)

= 2.9118+1.5731

⇒ logx = 4.4849

⇒ x = antilog 4.4849 = 30542

Let x = 816.3 ÷ 37.42

Taking log on both side we get,

⇒ logx = log (816.3 ÷ 37.42)

= log 816.3 – log 37.42 (since, log a ÷ b = log a – log b)

= 2.9118–1.5731

⇒ logx = 1.3387

⇒ x = antilog 1.3387 = 21.812

Let x = 0.000645 × 82.3

Taking log on both side we get,

⇒ logx = log (0.000645 × 82.3)

= log 0.000645 +log 82.3 (since, log a × b = log a +log b)

= ̅3.1904 + 1.9153

= –3.1904+1.9153

=–1. 2751

⇒ logx = –1.2751 = ̅1 . 2751

⇒ x = antilog ̅1.2751 = 0.05307

Let x = 0.3421 ÷ 0.09782

Taking log on both side we get,

⇒ logx = log (0.3421 ÷ 0.09782)

= log 0.3421 – log 0.09782 (since, log a÷b = log a –log b)

= ̅0.4658 – ̅1.00957

= –0.04658 – (–1.00957)

= –0.04658 + 1.00957

=0.54377

⇒ logx = 0.54377

⇒ x = antilog 0.54377= 3.497

Let x = (50.49)⁵

Taking log on both side

⇒ log x = 5 log (50.49) (∵ log aⁿ = n loga)

= 5 × 1.7032

logx = 8.516

⇒ x = antilog 8.516 = 32810000

Let x = ∛561.4

Taking log on both side

(∵ log aⁿ = n loga)

logx = 0.9163

⇒ x = antilog 0.9163 = 8.247

Let

Taking log on both side we get,

= log (175.23 × 22.159) – log (1828.56)

(∵ log a÷ b = loga – log b)

= log 175.23 + log 22.159 – log 1828.56

(∵ log a×b = loga + log b)

= 2.2436 + 1.3455 – 3.2621

⇒ log x = 0.327

⇒ x = antilog 0.327 = 2.123

Let

Taking log on both side we get,

(∵ log a÷ b = loga – log b)

(∵ log a×b = loga + log b)

(since, log aⁿ = n log a)

= 0.4823 + 0.5725 – 0.833

⇒ log x = 0.2218

⇒ x = antilog 0.2218 = 1.666

Let

Taking log on both side

⇒ log x = log ( (76.23)³ × ∛1.928 ) – log ((42.75)⁵ × 0.04623)

(∵ log a÷ b = loga – log b)

⇒ log x = log (76.23)³ +log ∛1.928 – (log (42.75)⁵ +log 0.04623)

(∵ log a × b = loga + log b)

⇒ log x = log (76.23)³ +log ∛1.928 – log (42.75)⁵ –log 0.04623

(since, log aⁿ = n log a)

⇒ log x = 5.6463 + 0.0950 – 8.1545 + 1.3350

⇒ log x = –1.0782 = ̅1.0782

⇒ x = antilog ̅1.0782 = 0.08352

Let

Taking log on both side,

(since, log aⁿ = n log a)

(∵ log a÷ b = loga – log b)

(∵ log a × b = loga + log b)

⇒ log x = –0.2255

⇒ x = antilog (–0.2255) = antilog ̅0.2255 = 0.5948

Let log₉ 63.28 = log₁₀63.28 × log₉10

(since, log_aM = log_bM × log_ab)

Then

Taking log on both side

⇒ log x = log 1.8012 – log 0.9542

(∵ log a÷ b = loga – log b)

⇒ log x = 0.2555 – (–0.0203)

= 0.2555+0.0203

= 0.2758

⇒ x = antilog 0.2758 = 1.887

Let log₃ 7 = log₁₀7× log₃10

(since, log_aM = log_bM × log_ab)

Then

Taking log on both side

⇒ log x = log 0.8450 – log 0.4771

(∵ log a÷ b = loga – log b)

⇒ log x = –0.0731– (–0.3213)

= – 0.0731 + 0.3213

= 0.2482

⇒ x = antilog 0.2482 = 1.771

Thus, 45₁₀ = 101101₂

Thus, 73₁₀ = 1001001₂

1101011₂ = 1 × 2⁶ + 1 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 0 × 2² +

1 × 2¹+ 1 × 2⁰

= 64 + 32+ 0+8+0+2+1 = 107₁₀

Thus, 1101011₂= 107₁₀

111₂ = 1 × 2² + 1 × 2¹ +1 × 2⁰

= 4 + 2+1 = 7₁₀

Thus, 111₂=7₁₀

Thus, 987₁₀ = 12422₅

Thus, 1238₁₀ = 14423₅

10234₅ = 1 × 5⁴ + 0 × 5³ + 2 × 5² + 3 × 5¹+ 4 × 5⁰

= 625 + 0 + 50+15+4 = 694₁₀

Thus, 10234₅ = 694₁₀

211423₅ = 2 × 5⁵ + 1 × 5⁴ + 1 × 5³ + 4 × 5² + 2 × 5¹+

3 × 5⁰

= 6250 + 625+ 125+100+10+3 = 7113₁₀

Thus, 211423₅ = 7113₁₀

Thus, 98567₁₀ = 300407₈

Thus, 688₁₀ = 1260₈

47156₈ = 4 × 8⁴ + 7 × 8³ + 1 × 8² + 5 × 8¹ + 6 × 8⁰

= 16384+3584+64+40+6 = 20078₁₀

Thus, 47156₈ = 20078₁₀

Thus, 585₁₀ = 1001001001₂

Thus, 585₁₀ = 4320₅

Thus, 585₁₀ = 1111₈

Let N = 923.4

Divide N by 10 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 923.4 = 9.234 × 10²

Correct answer is (B)

Let N = 0.00036

Divide N by 10⁵ to remove decimal, we get

Multiply and Divide N by 10, we get

Thus, scientific notation of 0.00036= 3.6 × 10^–4

Correct answer is (C)

2.57 x 10³ = 2.57 × 1000 = 2570

Correct answer is (B)

Correct answer is (A)

We know that x = log_ab is the logarithmic form of the exponential form b = a^x

Thus, here exponential form 5² = 25 is given

Where b = 25, a =5, x =2

Thus, its logarithmic form is 2 = log₅ 25

Hence, correct answer is (C)

We know that x = log_ab is the logarithmic form of the exponential form b = a^x

Thus, here logarithmic form log₂16 = 4 is given

Where b = 16, a =2, x =4

Thus, its logarithmic form is 2⁴ = 16

Hence, correct answer is (A)

Ans. Let

Thus, its exponential form is

On equating power of the base we get,

⇒ x = –1

Thus, correct answer is (D)

Let x = log₄₉7

Thus, its exponential form is

⇒ 49^x = 7

⇒ (7²)^x = 7

⇒ 7^2x = 7

On equating power of the base 7 we get,

⇒ 2x = 1

Thus, correct answer is (B)

Let

Thus, its exponential form is

On equating power of the base we get,

⇒ x = –2

Thus, correct answer is (A)

Consider, log₁₀8 + log₁₀5– log₁₀4 = log₁₀(8× 5) – log₁₀4

(since, log_aM + log_aN = log_a(M× N) )

⇒ log₁₀8 + log₁₀5– log₁₀4 = log₁₀(40) – log₁₀4

= log₁₀(40 ÷ 4)

(since, log_aM – log_aN = log_a(M÷N) )

⇒ log₁₀8 + log₁₀5– log₁₀4 = log₁₀(10) = 1 (since, log_aa = 1)

Thus, correct answer is (C)