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?
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 4: z will grow linearly, while y will not
ANSWER
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
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 log_{2}x 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:
Answer
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 = 3x1 ]
= a constant quantity
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 = 3x1 ]
= a constant quantity
Ques 3. f(x) grows exponentially with x, how will f(log(x)) grow?
Op 1: Exponentially
Op 2: Linearly
Op 3: Quadratically
Op 4: None of these
Answers:
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)
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 log_{512} 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 log_{7} (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 log_{64} 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: log_{7}85
Op 2: log_{85}7
Op 3: log_{10}7
Op 4: log_{10}85
Op 5:
Correct Op : 1
Ques 8. If log_{10}2 = 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(2^{64})

= 64 x log 2

= (64 x 0.30103)
 
= 19.26592

Its characteristic is 19.
Hence, then number of digits in 2^{64} is 20.
Ques 8. What is log_{1}10?
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 changeofbase formula. So, by the formula:
log[1](10) = ln(10) / ln(1) = ln(10) / 0
which doesn't exist
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 changeofbase formula. So, by the formula:
log[1](10) = ln(10) / ln(1) = ln(10) / 0
which doesn't exist
Ques 9. What is log_{10}0 ?
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 log_{3} (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.
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
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
Op 1: 2
Op 2: 2
Op 3: 0
Op 4: 4
Op 5:
Correct Op : 3
Ques 16. What is the value of log_{3} 1.5 + log_{3} 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 log_{8} x equivalent to?
Op 1: log_{2} (x/3)
Op 2: log_{2} (3x)
Op 3: (log_{2}x)/ 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
Op3:Quadratically
Op4:Noneofthese
Op 5:
Correct Op : 4
Op 1: 0
Op 2: 1
Op 3: 20
Op 4: None of these
Op 5:
Correct Op : 1
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