AMCAT Previous Years Question from Log with Solutions

AMCAT Previous Years Question from Log – Logarithms with Solutions

ALL PREVIOUS LOG QUESTIONS AND ANSWERS
Ques1. Every time x is increased by a given constant number, y doubles and z becomes three times. How will log(y) and log(z) behave as x is increased by the same constant number?
Op 1: Both will grow linearly with different slopes
Op 2: Both will grow linearly with same slopes
Op 3: y will grow linearly,
while z will not
Op 4: z will grow linearly, while y will not
Correct Op : 1
SOLUTION
y₂/y₁ = 2
log(y₂/y₁) = log(2)
log(y₂) – log(y₁) = log(2) = constant
log(y) increases linearly —> ∆log(y)/∆x = log(2)/k

z₂/z₁ = 3
log(z₂/z₁) = log(3)
log(z₂) – log(z₁) = log(3) = constant
log(z) increases linearly —> ∆log(z)/∆x = log(3)/k

Ques2. x triples every second. How will log2x change every second?
Op 1: It will double every second
Op 2: It will triple every second
Op 3: It increases by a constant amount every second.
Op 4: None of these
Op 5:
Correct Op : 3
SOLUTION
Let y = log(2) x
and let x change from x1 to x2.
If the corresponding values of y are y1 and y2
=> y1 = log(2) x1
and y2 = log(2) x2
Change in y = y2 – y1
= log(2) x2 – log(2) x1
= log(2) (x2/x1)
= log(2) 3 [ as per the question x2 = 3×1 ]
= a constant quantity
Ques 3. f(x) grows exponentially with x, how will f(log(x)) grow?
Op 1: Exponentially
Op 2: Linearly
Op 4: None of these
Correct:2
SOLUTION
f(x) = ab^x is the general exponential equation
log f(x) = log(a) + x*log(b)
which is in the form mx +c with m = log(b) and c = log(a)

Ques 4. What is the value of log512 8?
Op 1: 3
Op 2: 1/3
Op 3: -3
Op 4: -1/3
Op 5:
Correct Op : 2
SOLUTION

convert to exponential form
8^x=512=8^3
x=3
The logarithm of 512 to the base 8=-3
Ques 5. What is the value of log7 (1/49)?
Op 1: 2
Op 2: 1/2
Op 3: -1/2
Op 4: -2
Op 5:
Correct Op : 4
Solution :
WE HAVE 7^(-2)=1/49. therefore answer is -2
Ques 6. Given that log64 x = 2/6, what is the value of x? Op 1: 2
Op 2: 4
Op 3: 6
Op 4: 8
Op 5:
Correct Op : 2
Ques 7. If 7x = 85, what is the value of x?
Op 1: log785
Op 2: log857
Op 3: log107
Op 4: log1085
Op 5:
Correct Op : 1

Ques 8. If log102 = 0.3010, what is the number of digits in 264
Op 1: 19
Op 2: 20
Op 3: 18
Op 4: None of     these
Correct Op : 2
Solution:
 log(264) = 64 x log 2 = (64 x 0.30103) = 19.26592
Its characteristic is 19.
Hence, then number of digits in 264 is 20.
Ques 8. What is log110?
Op 1: 1
Op 2: 10
Op 3: 0
Op 4: Tends to infinity
Op 5:
Correct Op : 4
Solution:
Base 1 logarithms don’t exist. If there were such a value as log[1](10), then it would (by definition of a log) be the unique solution to:

1^x = 10

But this has no solution, so log[1](10) does not exist.

Alternatively, you can think of it in terms of the change-of-base formula. So, by the formula:

log[1](10) = ln(10) / ln(1) = ln(10) / 0

which doesn’t exist

Ques 9. What is log100 ?
Op 1: 0
Op 2: 10
Op 3: 1
Op 4: Not defined
Op 5:
Correct Op : 4
Solution:
As the graph for y=logx will approach but never reach x=0, the value for log0 is undefined. However, it does APPROACH negative infinity
Ques 10. What is the value of log3 (-9)?
Op 1: 3
Op 2: 1/3
Op 3: -3
Op 4: Not defined
Op 5:
Correct Op : 4
Solution:
log3 (-9)
Split -9 into 9*(-1).
=> log3(9)(-1)
=>log3(3^2)(-1)
Logarithmic rule says the power inside the bracket will come in front of log.
=> 2log3(3)(-1)
Another logarithmic rule says that when the base and the number are same then the log term cancels out.
=> 2(-1)
=> -2.

Thus , the ans is -2.

Ques 11. Rajeev multiplies a number by 10, the log (to base 10) of this number will change in what way?
Op 1: Increase by 10
Op 2: Increase by 1 Op 3: Multiplied by 10 Op 4: None of these Op 5:
Correct Op : 2
Ques 12. The logarithm of a very small positive number will tend to which of the following?
Op 1: 0
Op2:negativeinfiniyOp3:positiveinfinity Op 4: 1
Op 5:
Correct Op : 2
Ques 13. If n numbers are in geometric progression, the logarithm of the number will be in which of the following? Op 1: Geometric Progression
Op 2: Arithmetic Progression
Op 3: Harmonic Progression Op 4: None of these
Op 5:
Correct Op : 2
Ques 14. Which of the following is equivalent to log(a + b) ?
Op 1: log a + log b
Op 2: log a * log b
Op 3: log a – log b
Op 4: None of these
Op 5:
Correct Op : 4
Solution:
There are five rules for logarithms:

First: log a + log b = log (ab)
Example: log 2 + log 3 = log (2 times 3)
log 2 + log 3 = log 6
It is not equivalent to (log 2)(log 3) or (log 5)
Look at those brackets carefully.

Second: log a – log b = log (a/b)
Example: log 6 – log 2 = log (6/2)
log 6 – log 2 = log 3

log 1 = 0

log 0 = undefined

log (base a) a = 1

Ques 15. What is the value of log3 (1/9) + log 9 81 ?
Op 1: 2
Op 2: -2
Op 3: 0
Op 4: 4
Op 5:
Correct Op : 3
Ques 16. What is the value of log3 1.5 + log3 6 ?
Op 1: 2
Op 2: 2.7
Op 3: 1.8
Op 4: None of these
Op 5:
Correct Op : 1
Ques 17. Which of the following is log8 x equivalent to?
Op 1: log2 (x/3)
Op 2: log2 (3x)
Op 3: (log2x)/ 3
Op 4: None of these
Op 5:
Correct Op : 3
Ques 18. If n numbers are in arithmetic progression, the logarithm of the number will be in which of the following?
Op 1: Exponentially
Op 2: Linearly