Test: Logarithms (April 11) - JEE MCQ

According to the given options
(1) log₁₀10 = 1
Since, log_aa = 1
So, log₁₀10 = 1
It is correct.
(2) log (2 + 3) = log (2 × 3)
⇒ log(2 + 3) = 5
And, log(2 × 3) = log 6
⇒ log 2 + log 3
So, log (2 + 3) ≠  log (2 × 3)
It is not correct.
(3) log101 = 0
Since, log_a1 = 0
So, log₁₀1 = 0
It is correct.
(4) log (1 + 2 + 3) = log 1 + log 2 + log 3
⇒ log(1 + 2 + 3) = log 6
⇒ log(1 × 2 × 3)
⇒ log1 + log2 + log3
So, It is correct.
∴ The required option (2) is correct.

Given:
log₂(x + 1) = 2
Calculation:
⇒ log₂(x + 1) = 2
⇒ log₂(x + 1) = 2 × log22 = log₂2²
⇒ (x +1) = 4
⇒ x = 3
∴The answer is 3 .

Calculation:
log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7
⇒ (log₃2 ⋅ log₄3)  (log₅4 ⋅ log₆5)  (log₇6 ⋅ log₈7)
[log_bM × log_ab = log_aM]
⇒ log₄2 .log₆4. log₈6
⇒ (log₄2 .log₆4) log₈6
⇒ log₆2 ⋅ log₈6
⇒ log₈2
⇒ 1/log₂8        [∵log22=1]
⇒ 1/log₂2³ = 1/3log₂2 = 1/3​        
∴  log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7 = 1/3

Given:
log₂₇x = 1/6
Concept used:
If Log_ex = z, then x = e^z
Calculation:
log₂₇x = 1/6
⇒ x = 27^1/6 = √3
∴The answer is √3.

Formula used:
a^logax = x
Calculation:
7^log7(x²−4x+5) = x−1
By using the above property
⇒ x² - 4x + 5 = x -1
⇒ x² - 5x + 6 = 0
⇒ x² - 2x - 3x + 6 = 0
⇒ x(x - 2) - 3(x - 2) = 0
⇒ (x - 2)(x - 3) = 0
∴ x = 2 & 3

Logarithm properties:  
Product rule: The log of a product equals the sum of two logs.
log_a(mn) = log_am+log_an
Quotient rule: The log of a quotient equals the difference of two logs.
log_am/n = log_am−log_an
Power rule: In the log of power the exponent becomes a coefficient.
log_amⁿ = nlog_am
Formula of Logarithms:
If log_ax = b then x = a^b (Here a ≠ 1 and a > 0)

Concept:
a^b = x ⇔ log_ax = b, where a ≠ 1 and a > 0 and x be any number.
Calculation:
Given: 92^1/5 = 4.
As we know that, a^b = x ⇔ log_ax = b.
Comparing 92^1/5 = 4 with a^b = x we have,
Here, a = 92, b = 1 / 5 and x = 4.
So, the logarithmic form of 92^1/5 = 4 is log₉₂4 = 1/5.

Calculation

Logarithm properties:  
Product rule: The log of a product equals the sum of two logs.
log_a(mn) = log_am+log_an
Quotient rule: The log of a quotient equals the difference of two logs.
log_a⁽m/n)  = log_am−log_an
Power rule: In the log of power the exponent becomes a coefficient.
log_amⁿ = nlog_am
Formula of Logarithms:
If log_ax = b then x = a^b (Here a ≠ 1 and a > 0)

We have 5^x-1 = (2.5)^log10⁵
⇒ (2.5)log10⁵ = 5^x-1 
⇒ log₁₀⁵  = log_2.5^5x-1 
⇒ log105  = (x - 1) log2.55
⇒ (x - 1) = (log105)/(log2.55)
⇒ (x - 1) = log₁₀^2.5
⇒ x = log102.5 _{+ 1}
⇒ x = log₁₀2.5 ₊ log₁₀¹⁰
⇒ x = log₁₀¹⁰ × 2.5
⇒ x = log₁₀²⁵
⇒ x = log₁₀5²
⇒ x = 2log_10​5
∴ The value of x is 2log₁₀​5.