Problems on Logarithm

Problems on Logarithm

Learn and practice Problems on logarithm with easy explaination and shortcut tricks. All questions and answers on logarithm covered for various Competitive Exams.

Solve Problems:

1) Find the logarithm of 1/256 to the base 2√2.

  1. 16
  2. 13/5
  3. -16/3
  4. 12
View Answer

Answer: C

Explanation:

Let log2√2 [1/256] = x

We know that loga y = x is similar to ax = y

So, we can write it as [1/256] = (2√2) x

Or, (2√2) x = [1/28]

Or, [21 * 21/2]x = 1/28

Or, 23x/2 = 2-8

Therefore, 3x/2 = -8

Hence, x = (-8 * 2)/ 3 = -16/3

2) If loga [1/36] = -2/3, find the value of a.

  1. 6
  2. 8
  3. 9
  4. 216
View Answer

Answer: D

Explanation:

ATQ, loga [1/36] = -2/3

We know that loga y = x is similar to ax = y

So, a (-2/3) = 1/36

Or, a (-2/3) = 1/62

Or, a (-2/3) = 6-2

Now, multiply and divide by 3 in the power of 6 to make the power equals to power of a.

So, a (-2/3) = 63(-2/3)

On comparing both side, we get a = 63

Therefore, a = 216

3) Find the value of x

Log4(log8 64) = log5 x

  1. 2
  2. 6
  3. 5
  4. √5
View Answer

Answer: D

Explanation:

ATQ, Log4 (log8 64) = log5 x…… (i)

Let, log8 64 = a

Or, 64 = 8a, or 8a = 82

That means a=2

Now, put log8 64 = 2in equation 1.

Log4 (2) = log5 x…… (ii)

Now, let log4 2 = s

Or, 4s = 2, or 22S = 2

Now, on comparing both side we get 2s = 1, or s = ½

Put the value of s in equation 2

So, log5 x = ½

Therefore, x = 51/2 = √5

4) The equation, a2 + b2 = 7ab equals to

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View Answer

Answer: A

Explanation:

Here, a2 + b2 = 7ab

Add both sides 2ab to make a formula.

Or, a2 + b2 + 2ab = 7ab + 2ab

Or, (a+b) 2 = 9ab

Or, (a+b) 2 / 9 = ab

Or, [1/3 (a+b)]2 = ab

Now, taking log both sides

Therefore, log [1/3 (a+b)]2 = log ab

We know that log m*n = log m + log n, and log mn = n log m

So, 2 log [1/3 (a+b)] = log a + log b

Therefore, log [1/3 (a+b)] = 1/2(log a + log b)

= a2 + b2

= 7ab

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5) If (log3 x)(logx 2x)(log2x y) = logx x2, find y.

  1. 6
  2. 9
  3. 5
  4. 7
View Answer

Answer: A

Explanation:

(log3 x)(logx 2x)(log2x y) = logx x2
We know that logb a =
Now, A =

B =

C =

Now, ATQ, A* B * C = logx x2
Or, ** =

We know that, loga mn = n loga m
=

We know that loga a = 1
That means, log3 y = 2
We know thatloga y = x is similar to ax = y
Therefore, y = 32 = 9

6) Add the equation x = 1 + loga bc, y = 1 + logb ca and z = 1 + logc ab. Find the value of 

  1. xyz = xy + yz + zx
  2. X2yz = xyz + yx + 1
  3. (xz + 1)2 = xy – yz – zx
  4. (xz + y) = xy + yz + zx
View Answer

Answer: A

Explanation:

x =1 + loga bc

We can write it as x = loga a + loga bc => loga abc

Similarly, y = logb b + logb ca => logb abc

And, z = logc c + logc ab => logc abc
Changing the values of x, y, z to a common base m by using logb a = we get
X =

Y =

Z =

Now, =[logm a + logm b + logm c]

Now, Right hand side equation, [logm abc/ logm abc] = 1
Now, = 1

Or, = 1 => xyz =xy + yz + zx.

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7) Find the characteristics of the logarithms of i) 5631, ii) 5.678, iii) 56.23.

  1. 3, 1, 5
  2. 3, 0, 1
  3. 6, 5, 9
  4. 8, 7, 6
View Answer

Answer: B

Explanation:

i) The number 5631> 1 and the number of digits in the integral part are 4.
Therefore, the characteristics of the logarithm are 4-1 = 3,

ii) In the number 5.678, the number of digits in the integral part = 1.
Therefore, characteristics of the logarithm are 1-1 =0

iii) In this case, the number of digits in an integral part of the number 56.23 are 2.
Therefore, the characteristics of the logarithm = 2-1 = 1

So, the option b is correct.

8) If log 2 = 0.30103, log 3 = 0.47712, the number of digits in 620 is

  1. 8
  2. 12
  3. 16
  4. 20
View Answer

Answer: C

Explanation:

We have to take 620 with log
We know that log mn = n log m
Now, log 620 = 20 log 6
Or, 20 log (2*3)
We know that log (m*n) = log m + log n
Now, 20 [log 2 + log 3] = 20 [0.30103 + 0.47712] = 20 [0.77815]
Since, the number of digits in 620 = 15.563, round up value = 16
Therefore the number of digits = 16

9) The logarithm of 0.0001 to the base 0.001 is equal to

  1. 4/3
  2. 5/3
  3. 7/3
  4. 2/3
View Answer

Answer: A

Explanation:

Let Log0.001 (0.0001) = x

We know that logx y = a equals to xa = y.

So, (0.001)x= 0.0001
Or, = [1/10000]

Or, =

Therefore, compare both side, we get 3x = 4.

Hence, x = 4/3

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10) If logb x = ͞5.1342618, then the value of log10(x(1/4)) will be

  1. ͞1.2835655
  2. ͞2.7164345
  3. ͞2.7835655
  4. ͞3.2164345
View Answer

Answer: C

Explanation:

Here logb x = ͞5.1342618 is given and log10(x(1/4)) is asked

That means here b also treat as base 10

Now, we can say log10 x = ͞5.1342618 = -5 + 0.1342618 => -4.8657382

Therefore, log10 (x1/4) = ¼ log10 x

Or, ¼ (-4.8657382) = -1.21643455, but it is not in the option.

To make it -2, we have to subtract 0.7835655 from -1.21643455

i.e., -1.21643455 – 0.7835655 = -2

Now, to get back on the original value we have to add 0.7835655 in -2.

Or, -2 + 0.7835655 = ͞2.7835655

Hence, the c is correct.

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