# Cognizant Logarithms Questions with Solutions

1. If ax = by, then:

A: log(a/b) = x/y
B: log(a) / log(b) = x/y
C: log(a) / log(b) = y/x
D: None of these

1. If log x + log y = log (x + y), then:

A: x = y
B: xy=1
C: y = (x-1)/x
D: y = x/(x-1)
Explanation: log x + log y = log (x+y)
log(xy) = log (x+y)xy = x + y
xy – y = x
(x-1)y = x y = x/(x-1)

1. If log10 7 = a, then log10(1/70) is equal to:

A: -(1 + a)
B: (1 + a)-1
C: a/10
D: 1/10a
Explanation: log10(1/70)=log10 1- log 10 70
= 0-log10 (10*7)
=-((log10 10 )+(log10 7))
=- (1+a)

1. If log{(a+b)/3} = 0.5(log a + log b), then the correct relation between a and b is:

A: a2+b2 = 7ab

B: a2-b2 = 7ab
C: (a+b)2 = 2
D: (a+b)/3 = (1/2)(a+b)
Explanation:
Log[(a+b)/3] = 0.5[log(a) + log(b)]
log[(a+b)/3] = 0.5*log(ab)
log[(a+b)/3] = log[sqrt(ab)]
(a+b)/3 = sqrt(ab)
a + b = 3*sqrt(ab)
a^2 + 2ab + b^2 = 9 ab
a^2 + b^2 = 7ab

1. If log x = log 3 + 2 log 2- (3/4) log 16. The value of x is:

A: ½
B: 1
C: 3/2
D: 2
Explanation: log x= log 3+ 2 log 2 – (3/4)log 16.
log x = log 3 + 2 log 2 – (3/4) log 2^4.
log x = log 3 + 2 log 2 – (3*4)/4 log 2.
log x = log 3 + 2 log 2 – 3 log 2.
log x = log 3 – log 2.
log x = log 3/2.
Therefore, by equating, x = 3/2.

1. If log x =(1/2) log y = (1/5) log z, the value of x4y3z-2 is:

A: 0
B: 1
C: 2
D: 3
log(x) = (1/2) log(y) => y = x^2
log(x) = (1/5) log(z) => z = x^5
Then x^4*y^5*z^-2 = x^4*x^6*x^-10 = x^10 * x^-10 = x^0 = 1,
So correct option is opn 2 i.e 1.

1. If log10000 x = -1/4, then x is given by:

A: 1/100
B: 1/10
C: 1/20
D: none of these
Log10000x=-1/4
X=(10000)-1/4
10-1=1/10

1. The value of 3-1/2 log3(9) is:

A: 3
B: 1/3
C: 2/3
D: none of these
3^{-1/2 log3(3^2)}
3^{-2/2 log3(3)}
3^{-1 log3(3)}
3^(-1)
It means 1/3

1. loge xy – loge |x| equals to:

A: loge x
B: loge |x|
C: – loge x
D: none of these
Explanation:
log x= log 3+ 2 log 2 – (3/4)log 16.
log x = log 3 + 2 log 2 – (3/4) log 2^4.
log x = log 3 + 2 log 2 – (3*4)/4 log 2.
log x = log 3 + 2 log 2 – 3 log 2.
log x = log 3 – log 2.
log x = log 3/2.

1. The value of (loga n) / (logab n) is given by:

A: 1 + loga b
B: 1 + logb a
C: loga b
D: logb a
Explanation:
log (base p) x = ln(x) / ln(p)
So, log (base a) x / log (base ab) x = ln(ab) / ln(a)
Since ln(ab) = ln a + ln b, then
ln(ab) / ln(a) = 1 + (ln a)/(ln b) = 1 + log (base b) a

1. If a, b, and c are in geometric progression then loga n, logb n and logc n are in:

A: AP
B: GP
C: HP
D: None of these
Explanation:
They are in H.P.
a, b and c are in GP ( b^2=ac)
b^2=ac
logb^2= log a + log c
{all log are To base n }
loga , log b and logc are in AP
so, 1/loga , 1/logb and 1/logc are in HP
applying base changing law of log
loga to base n = 1/ {logn to base a}
==> 1/ loga = log(base a ) n
similarly , 1/log( base b ) n
and 1/log (base c) n
then
(loga n , logb n ,logc n) are in H.P.

1. What is the value of antilog10100?

A: 2
B: 10100
C: 100
D: 10
Explanation:
Because log(10) 100 = 2, antilog(10) 2 = 100 or 10^2 = 100
The antilogarithm is the inverse function of a logarithm, so log(b) x = y means that antilog (b) y =x

1. If antilog x 5 = 30, what can you infer about x?

A: x is a number between 1 and 2
B: x is 305
C: x is a number between 2 and 3
D: None of these
X^5 = 30
1^5 = 1, 2^5 = 32
The inference is that
x is a number between 1 and 2

1. x triples every second. How will log2x change every second?

A: It will double every second
B: It will triple every second
C: It increases by a constant amount every second.
D: None of these
Explanation:
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

15 .  What is the value of log512 8?
A) 1: 3
B) 1/3
C) -3
D)1/3
Correct Op : 2
SOLUTION
convert to exponential form
8^x=512=8^3
x=3
The logarithm of 512 to the base 8=-3

1. What is the value of log7 (1/49)?

A) 2
B) 1/
C) ½
D )2:
Correct Op : 4
Solution :
WE HAVE 7^(-2)=1/49. therefore answer is -2

1. Ques 8. What is log110?

A) 1
B) 10
C) 0
D) Tends to infinity
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

1. What is log100 ?

A) 0
B) 10
C) 1
D) Not defined
Correct Op : D
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

1. Given that log64 x = 2/6, what is the value of x?

A: 2
B: 4
C: 6
D: 8