Surds, Indices and Logarithm

log_x625 = 4
Or, x⁴ = 625 = 5⁴
Or, x = 5

দেওয়া আছে,
4^{x + 1} = 32
⇒ 2^{2(x + 1)} = 2⁵
⇒ 2(x + 1) = 5
⇒ 2x + 2 = 5
⇒ 2x = 5 - 2
⇒ 2x = 3
∴ x = 3/2

Question: If (a/b)^{3x - 4} = (b/a)^{x + 2}, then what is the value of x?

Solution:
Given, (a/b)^{3x - 4} = (b/a)^{x + 2}
⇒ (a/b)^{3x - 4} = (a/b)^{- (x + 2)}
⇒ (a/b)^{3x - 4} = (a/b)^{- x - 2}
⇒ 3x - 4 = - x - 2
⇒ 3x + x = - 2 + 4
⇒ 4x = 2
∴ x = 1/2

Question: Find the value of x if log_x 256 = 4.

Solution:
log_x 256 = 4
⇒ x⁴ = 256
⇒ (x²)² = 256
⇒ x² = 16
⇒ x = √16
⇒ x = 4

∴ The value of x is 4.

Question:

Solution:

Question: If 9^√x + 9^√x = 162 then, what is the value of x?

Solution: 
Given that, 
9^√x + 9^√x = 162
⇒ 9^√x(1 + 1) = 162
⇒ 9^√x × 2 = 162
⇒ 9^√x = 162/2
⇒ 9^√x = 81
⇒ 9^√x = 9²
⇒ √x = 2
⇒ (√x)² = 2²
∴ x = 4

Given. xyz=1,a^x=b,b^y=c
Now, b=a^x
=> b^y=a^xy
=> b^yz=a^xyz
=> c^z=a

Question:

Solution:

irrational unless n is the mth power of an integer. If m and n are natural numbers, then ^m√n is irrational unless n is mth power of an integer

Question: If 2^{n - 1} + 2^{n + 1} = 320, then the value of n is = ?

Solution:
Given that, 
2^{n - 1} + 2^{n + 1} = 320
⇒ 2^{n - 1} + 2^{n - 1} . 2² = 320
⇒ 2^{n - 1}(1 + 2²) = 320
⇒ 2^{n - 1} . 5 = 320
⇒ 2^{n - 1} = 320/5 = 64
⇒ 2^{n - 1} = 26
⇒ n - 1 = 6
⇒ n = 6 + 1
∴ n = 7

So the value of n is 7.

Question: If an exponent or index has base 15 and power zero, then which of the following will be its value?

Solution:
a⁰ = 1 (for any non-zero base a)

Now,
= 15⁰
= 1

Question: If log₄[log₃(log₂x)] = 0, then find the value of x.

Solution:
log₄[log₃(log₂x)] = 0
⇒ log₃(log₂x) = 4⁰  [log_bM = c ⇒ M = b^c]
⇒ log₃(log₂x) = 1
⇒ log₂x = 3¹  [log_bM = c ⇒ M = b^c]
⇒ log₂x = 3
⇒ x = 2³  [log_bM = c ⇒ M = b^c]
∴ x = 8

Question: 


Solution:

Question: If log₃(x² - 4x) - log₃(x - 4) = 4, Than what is the value of x. 

Solution:
Given that, 
log₃(x² - 4x) - log₃(x - 4) = 4
⇒ log₃[x(x - 4)/(x - 4)] = 4     ; [log_aM - log_aN = log_a(M/N)]
⇒ log₃x = 4
⇒ x = 3⁴
∴ x = 81

Question: If x and y are positive integers such that 3^{x + y} = 9⁴ and 2^{x - y} = 16, what is the value of x² - y²?

Solution: 
Given that, 
3^{x + y} = 9⁴
⇒ 3^{x + y} = (3²)⁴
⇒ 3^{x + y} = 3⁸
∴ x + y = 8 ........(1)

And,
2^{x - y} = 16
⇒ 2^{x - y} = 2⁴
∴ x - y = 4 ....... (2)

Now (1) + (2) than we get,
⇒ (x + y) + (x - y) = 8 + 4
⇒ 2x = 12
⇒ x = 12/2
∴ x = 6

From (1),
6 + y = 8
⇒ y = 8 - 6
∴ y = 2

∴ x² - y² = (6)² - (2)²
= 36 - 4
= 32

Question: If 3^{x + y} = 81 and 3^{x - y} = 9, then what are the values of x and y respectively?

Solution:
Given,
3^{x + y} = 81
⇒ 3^{x + y} = 3⁴
⇒ x + y = 4 .......(1)

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

Now, solving (1) and (2) we get,
x + y + x - y = 4 + 2
⇒ 2x = 6
⇒ x = 3

Now, x + y = 4
⇒ 3 + y = 4
⇒ y = 4 - 3
⇒ y = 1

∴ (x, y) = (3, 1)

Question: If √(2ⁿ) = 64, then the value of n is-

Solution: 
Given that, 
√(2ⁿ) = 64
⇒ (2ⁿ)^1/2 = 2⁶
⇒ 2^n/2 = 2⁶
⇒ n/2 = 6
⇒ n = 2 × 6
∴ n = 12

Question:

Solution:

Question:

Solution: 

Question: Find the value of x, if 9^{2x + 1} = 81³.

Solution:
9^{2x + 1} = 81³
⇒ (3²)^{2x + 1} = (3⁴)³
⇒ 3^{2(2x + 1)} = 3¹²
⇒ 4x + 2 = 12
⇒ 4x = 10
⇒ x = 10/4
∴ x = 2.5

Question: Given,

Solution: 

প্রশ্ন: যদি log_x(1/125) = - 3 হয়, তবে x এর মান কত?

সমাধান:
log_x(1/125) = - 3
⇒ x^{- 3} = 1/125    [log_a(b) = c  ⇒  a^c = b]
⇒ 1/x³ = 1/125
⇒ x³ = 125
∴ x = 5

Question: 2log₁₀5 + log₁₀8 - (1/2)log₁₀4 = ?

Solution:

Question: Solve for x: log₃(x + 5) = 3

Solution:
Given that,
log₃(x + 5) = 3
⇒ x + 5 = 3³  [log_ab = c ⇒ b = a^c] 
⇒ x + 5 = 27
⇒ x = 27 - 5
∴ x = 22

log_x9/16 = – 1/2 
⇒ x^-1/2 = 9/16
⇒ √x = 16/9
⇒ x = (16/9)²   
∴ x = 256/81

Question: If 7^b = 343, then the value of 7^{(b - 2)} is:

Solution:
Given that,
7^b = 343
⇒ 7^b = 7³
∴ b = 3

Now,
7^{(b - 2)}
= 7^{(3 - 2)}
= 7¹
= 7 

log_x125 = 3
Or, x³ = 125 = 5³
Or, x = 5

Question: If 2^a = 3^b and 2^{a + 2} - 3^{b + 1} = √3, then find the value of b.

Solution: 
Given, 
2^{a + 2} - 3^{b + 1} = √3
⇒ (2^a × 2²) - (3^b × 3¹) = √3
⇒ (3^b × 4) - (3^b × 3) = √3
⇒ 3^b(4 - 3) = √3
⇒ 3^b = 3^1/2
∴ b = 1/2

Question: What is the value of the following expression?

Solution:

= log₆₀ 3 + log₆₀ 4 + log₆₀ 5 [∵ 1/log_ba = log_ab]
= log₆₀(3 × 4 × 5) [∵ log_b(m) + log_b(n) = log_b(m × n)]
= log₆₀ 60
= 1

Question:

Solution:

Question: If x and y are whole numbers such that x^y = 64, then the value of (x - 2)^{y + 1} is = ?

Solution:
দেওয়া আছে,
x^y = 64
⇒ x^y = 8²
এখানে, x = 8 এবং y = 2

এখন,
(x - 2)^{y + 1} = (8 - 2)^{2 + 1}  [মান বসিয়ে]
= 6³
= 216

 Here, log_x50 = log_x(2×25)
= log_x2 + log_x5²
= log_x2 + 2log_x5
= a + 2b [As, log_x2 = a; log_x5 = b]

Question: If log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1, then what is the value of x ?

Solution:
log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + 1
⇒ log₁₀5+ log₁₀(5x + 1) = log₁₀(x + 5) + log₁₀10
⇒ log₁₀[5(5x + 1)] = log₁₀[10(x + 5)
⇒ 5(5x + 1) = 10(x + 5)
⇒ 5x + 1 = 2x + 10
⇒ 3x = 9
∴ x = 3

Question:

Solution: