- 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

Correct Answer : C

- 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)

Correct Answer : D

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)

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

A: -(1 + a)

B: (1 + a)-1

C: a/10

D: 1/10a

Correct Answer : A

Explanation: log10(1/70)=log10 1- log 10 70

= 0-log10 (10*7)

=-((log10 10 )+(log10 7))

=- (1+a)

- 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)

Correct Answer : A

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

- 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

Correct Answer : C

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.

- 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

Correct Answer : B

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.

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

A: 1/100

B: 1/10

C: 1/20

D: none of these

Correct Answer : B

Log10000^{x}=-1/4

X=(10000)^{-1/4}

10^{-1}=1/10

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

A: 3

B: 1/3

C: 2/3

D: none of these

Correct Answer : B

3^{-1/2 log3(3^2)}

3^{-2/2 log3(3)}

3^{-1 log3(3)}

3^(-1)

It means 1/3

- loge xy – loge |x| equals to:

A: loge x

B: loge |x|

C: – loge x

D: none of these

Correct Answer : D

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.

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

A: 1 + loga b

B: 1 + logb a

C: loga b

D: logb a

Correct Answer : b

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

- 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

Correct Answer : C

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.

- What is the value of antilog10100?

A: 2

B: 10100

C: 100

D: 10

Correct Answer : B

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

- 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

Correct Answer : A

X^5 = 30

1^5 = 1, 2^5 = 32

The inference is that

x is a number between 1 and 2

- 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

Correct Answer : C

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 log_{512} 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

- What is the value of log
_{7}(1/49)?

A) 2

B) 1/

C) ½

D )2:

Correct Op : 4

Solution :

WE HAVE 7^(-2)=1/49. therefore answer is -2

- Ques 8. What is log
_{1}10?

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

- What is log
_{10}0 ?

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

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

A: 2

B: 4

C: 6

D: 8

Correct Answer : B

Explanation: The answer is 4 can written as 4 cube

- If log102 = 0.3010, what is the number of digits in 264

A: 19

B: 20

C: 18

D: None of these

Correct Answer : B

Explanation:

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