Surds, Indices and Logarithm

Question: If e^x = 7, then x =?

Solution: 
Given, 
e^x = 7
⇒ ln(e^x) = ln 7
∴ x = ln 7;  [Formula: ln(e^x) = x]

3 coaches and 3 batsman or 2 bowlers and 4 coaches means
(3 coaches x 3 batsman) + (2 bowlers x 4 coaches)

Select 3 coaches out of 5 = ⁵C₃ = 5!/(3! × 2!) = 10

Select 3 batsman out of 4 = ⁴C₃ = 4!/(3! × 1!) = 4

Select 2 bowlers out of 3 = ³C₂ = 3!/(2! × 1!) = 3

Select 4 coaches out of 5 = ⁵C₄ = 5!/(4! × 1!) = 5

Total ways to form the group = (10 × 4) + (3 × 5) = 40 + 15 = 55.

Question: 


Solution:

(3⁴x¹⁷y¹⁸)^1/4
= (3^4/4x^17/4y^18/4)
= 3x^17/4y^9/2

Question: If a and b are whole numbers such that, a^b = 32; the value of (a + 1)^{2b - 7} is-

Solution:
Here, a^b = 32
a^b = 2⁵
∴ a = 2 and b = 5 

Now,
(2 + 1)^{(2 × 5) - 7} = 3^{(10 -7)}
= 3³
= 27

Given, (1/5)^3y = 0.008 = 8/1000
Or, (1/5)^3y = 1/125 = (1/5)³
Or, 3y = 3
Or, y = 1

So, (0.25)^y = (0.25)¹ = 0.25

Question: If a^x = b^y, then:

Solution:
a^x = b^y
⇒ log a^x = log b^y
⇒ x log a = y log b
⇒ (log a)/ (log b) = y/x

Question:

Solution:

Question: If x = 10^1.4, y = 10^0.7 and x^z = y³, then what is the value of z?

Solution:
Given,
x = 10^1.4, y = 10^0.7

Now,
x^z = y³
⇒ (10^1.4)^z = (10^0.7)³
⇒ 10^1.4z = 10^2.1
⇒ 1.4z = 2.1
⇒ z = 2.1/1.4
⇒ z  = (2.1 × 10)/(1.4 × 10)
⇒ z = 21/14
∴ z = 3/2

Question: Which of the following statements is not correct?

Solution:
Option ক)
Since log_aa = 1
So, log⁡₁₀10 = 1 
This is correct.

Option খ)
log⁡(2 + 3) = log⁡(2 × 3)
Compute the left side- 2 + 3 = 5, so log⁡(2 + 3) = log⁡5.
Compute the right side- 2 × 3 = 6, so log⁡(2 × 3) = log⁡6.
Logarithm property: log⁡(a⋅b) = log⁡a + log⁡b, not log⁡(a + b).
This is incorrect.

Option গ)
Since log_a1 = 0, so log₁₀1 = 0.
This is correct.

Option ঘ)
log (1 + 2 + 3) = log 1 + log 2 + log 3
Compute the left side- 1 + 2 + 3 = 6, so log⁡(1 + 2 + 3) = log⁡6.
Right side- log⁡1 + log⁡2 + log⁡3 = log(1 × 2 × 3) = log6
Both sides are equal: log⁡6 = log⁡6
This is correct.

Option খ) is the only statement that is not correct.

প্রশ্ন: প্রদত্ত 

সমাধান:

Question: If a and b are positive real numbers, then (a⁰ - 3b⁰)⁵ = ?

Solution:
We know that for any positive real number,
a⁰ = 1 and b⁰ = 1

So, (a⁰ - 3b⁰)⁵
= (1 - 3 × 1)⁵
= (1 - 3)⁵
= (- 2)⁵
= - 32

Question: If 3^{x + 3} + 7 = 250, then x is equal to?

Solution:
Given that, 
3^{x + 3} + 7 = 250
⇒ 3^{x + 3} = 250 - 7
⇒ 3^{x + 3} = 243
⇒ 3^{x + 3} = 3⁵
⇒ x + 3 = 5
⇒ x = 5 - 3
∴ x = 2

So the value of x is 2

প্রশ্ন: If log_x(81/16) = - 2, then x = ?

সমাধান:
দেওয়া আছে,
log_x(81/16) = - 2
⇒  x^{- 2} = 81/16
⇒  x^{- 2} = (9/4)²
⇒ x^-2 = (9/4)²
⇒  x^{- 2} = 1/(4/9)²
⇒  x^{- 2} = (4/9)^{- 2}
∴ x = 4/9

Question: If P = 216^{- 1/3} + 243^{- 2/5} + 256^{- 1/4}, then which one of the following is an integer?

Solution:
P = 216^{- 1/3} + 243^{- 2/5} + 256^{- 1/4}
= (6³)^{- 1/3} + (3⁵)^{- 2/5} + (4⁴)^{- 1/4
=} 6^{3(- 1/3)} + 3^{5(- 2/5)} + 4^{4(- 1/4)}
= 6^{- 1}+ 3^{- 2}+ 4^{- 1}
= (1/6)+ (1/9) + (1/4)
= (6 + 4 + 9)/36
∴ P = 19/36

Now,
Option (A): P/19 = (19/36)/19 = 1/36, not an integer. Reject.
Option (B): P/36 = (19/36)/36 = 19/36², not an integer. Reject.
Option (C): 36/P = 36/(19/36) = 36²/19, not an integer. Reject.
Option (D): 19/P = 19/(19/36) = 36, an integer. Correct.
Option (E): P = 19/36, not an integer. Reject.

Question: If log_m (1/√32) = - 5/2 what is the value of m?

সমাধান:
দেওয়া আছে,
log_m (1/√32) = - 5/2
⇒ m^{- 5/2} = 1/√32 [log_aM = x হলে, a^x = M হয়]
⇒ m^{- 5/2} = 1/(32^1/2)
⇒ m^{- 5/2} = 32^{- 1/2}
⇒ m^{- 5/2} = (2⁵)^{- 1/2}
⇒ m^{- 5/2} = 2^{- 5/2}
∴ m = 2

Question: If 9^{x + y} = 1 and 9^{x - y} = 3, then what are the values of x and y respectively?

Solution:
Given,
9^x+y = 1
⇒ 9^{x + y} = 9⁰
⇒ x + y = 0 .......(1)|

Again,
9^{x - y} = 3
⇒ 9^{x - y} = 3¹
⇒ (3²)^{x - y} = 3¹
⇒ 3^{2(x - y)} = 3¹
⇒ 2(x - y) = 1
⇒ x - y = 1/2 .............(2)

Now, solving (1) and (2) we get,
x + y = 0
x - y = 1/2
⇒ 2x = 1/2
∴ x = 1/4

Now,
x + y = 0
⇒ 1/4 + y = 0
⇒ y = - 1/4

Question: If log⁡_y81 = 4/2​, what is the value of y?

Solution: 
log⁡_y81 = 4/2
⇒ log⁡_y81 = 2
⇒ y² = 81  [log_ba = c ⇒ b^c = a]
⇒ y² = 9²
∴ y = 9

Question: If 4^a = 5, 5^b = 6, 6^c = 7, 7^d = 8, then the value of abcd is = ?

Solution:
8 = 7^d
= (6^c)^d
= 6^cd
= (5^b)^cd
= 5^bcd
= (4^a)^bcd
= 4^abcd
⇒ 4^abcd = 8
⇒ (2²)^abcd = 2³
⇒ 2abcd = 3
∴ abcd = 3/2

(10)²⁰⁰÷(10)¹⁹⁶
=(10)^200−196
= 10⁴
=10000

Question: 4 log 2 + log 7 =?

Solution: 
4 log 2 + log 7
= log 2⁴ + log 7
= log 16 + log 7
= log (16 × 7)
= log 112

Question: If log_xy = 100 and log₃x = 20; then the value of y is-

Solution:
Given,
log₃x = 20
∴ x = 3²⁰ 

And, log_xy = 100
⇒ y = x¹⁰⁰ 
⇒ y = (3²⁰)¹⁰⁰
∴ y = 3²⁰⁰⁰

3^x + 3^x + 3^x
= 3.3^x
= 3¹.3^x
= 3^{x + 1}

Let x=log2⁷
=> 2x=7
which is only possible for irrational number

Question: If x and y are positive real numbers, then (2x⁰ - 5y⁰)³ = ?

Solution:
We know that for any positive real number,
x⁰ = 1 and y⁰ = 1

So, (2x⁰ - 5y⁰)³
= (2 × 1 - 5 × 1)³
= (2 - 5)³
= (- 3)³
= - 27

3log2+2log3
=log2³+log3²
=log8+log9
=log(8×9)
=log72

2log6+log2
=log6²+log2
=log36+log2
=log72

3log2+2log3 / 2log6+log2
=log72 / log72
= 1

Question: Find the value of n, if 9^{n - 1} = 243.

Solution:
9^{n - 1} = 243
⇒ (3²)^{n - 1} = 3⁵
⇒ 3^{2(n - 1)} = 3⁵
⇒ 3^{2n - 2} = 3⁵
⇒ 2n - 2 = 5
⇒ 2n = 5 + 2
⇒ 2n = 7
⇒ n = 7/2
⇒ n = 3.5

Question: If log_x 1/216 = - 3, then x = ?

Solution:
Given that, 
log_x 1/216 = - 3
⇒ x^{- 3} = 1/216
⇒ 1/x³ = 1/216
⇒ x³ = 216
⇒ x³ = 6³
∴ x = 6

Question: Solve for x: log₂(x + 5) = 3.

Solution:
Given,
log₂(x + 5) = 3
⇒ x + 5 = 2³ [log_ax = b ⇒ x = a^b]
⇒ x + 5 = 8
⇒ x = 8 - 5
∴ x = 3

Question: 


Solution:

Question: (1/2)(logx + logy) will equal to log{(x + y)/2} if -

Solution: 
(1/2)(logx + logy) = log{(x + y)/2}
⇒ (1/2)log(xy) = log{(x + y)/2}
⇒ log(xy)^1/2 = log{(x + y)/2}
⇒ (xy)^1/2 = (x + y)/2
⇒ xy = {(x + y)/2}²
⇒ 4xy = x² + y² + 2xy
⇒ x² + y² - 2xy = 0
⇒ (x - y)² = 0
⇒ x - y = 0
∴ x = y