# LOGARITHM WORKSHEET

Logarithm Worksheet :

Worksheet given in this section will be much useful for the students who would like to practice problems on logarithms.

Before look at the worksheet, if you want to learn about logarithms in detail,

## Logarithm Worksheet - Problems

Problem 1 :

Find the logarithm of 64 to the base 2√2.

Problem 2 :

Find the value of log√264.

Problem 3 :

Find the value of log(0.0001) to the base 0.1.

Problem 4 :

Find the value of log (1/81) to the base 9.

Problem 5 :

Find the value of log(0.0625) to the base 2.

Problem 6 :

Find the value of log(0.3) to the base 9.

Problem 7 :

Given log2 = 0.3010 and log3 = 0.4771, find the value of log6.

Problem 8 :

If 2logx  =  4log3,  then find the value of 'x'.

Problem 9 :

If logabc  =  x, logbca  =  y and logcab  =  z, then find the value of

1/(x+1) + 1/(y+1) + 1/(z+1)

Problem 10:

If a = log2412, b = 2436 and c = log4836, then find the value of

1 + abc ## Logarithm Worksheet - Solutions

Problem 1 :

Find the logarithm of 64 to the base 2√2.

Solution :

Write 64 as in terms of 2√2.

64  =  26

64  =  24+2

64  =  2 22

64  =  2⋅ [(√2)2]2

64  =  2⋅ (√2)4

64  =  (2√2)4

Then,

log2√264  =  log2√2(2√2)4

log2√264  =  4log2√2(2√2)

log2√264  =  4(1)

log2√264  =  4

Problem 2 :

Find the value of log√264.

Solution :

log√264  =  log√2(2)6

log√264  =  6log√2(2)

log√264  =  6log√2(√2)2

log√264  =  6 ⋅ 2log√2(√2)

log√264  =  12 ⋅ 2(1)

log√264  =  12

Problem 3 :

Find the value of log(0.0001) to the base 0.1.

Solution :

log0.1(0.0001)  =  log0.1(0.1)4

log0.1(0.0001)  =  4log0.10.1

log0.1(0.0001)  =  4(1)

log0.1(0.0001)  =  4

Problem 4 :

Find the value of log (1/81) to the base 9.

Solution :

log9(1/81)  =  log91 - log981

log9(1/81)  =  0 - log9(9)2

log9(1/81)  =  -2log99

log9(1/81)  =  -2(1)

log9(1/81)  =  -2

Problem 5 :

Find the value of log(0.0625) to the base 2.

Solution :

log2(0.0625)  =  log2(0.5)4

log2(0.0625)  =  4log2(0.5)

log2(0.0625)  =  4log2(1/2)

log2(0.0625)  =  4(log21 - log22)

log2(0.0625)  =  4(0 - 1)

log2(0.0625)  =  4(-1)

log2(0.0625)  =  -4

Problem 6 :

Find the value of log(0.3) to the base 9.

Solution :

log9(0.3)  =  log9(1/3)

log9(0.3)  =  log91 - log93

log9(0.3)  =  0 - log93

log9(0.3)  =   - log93

log9(0.3)  =  - 1 / log39

log9(0.3)  =  - 1 / log332

log9(0.3)  =  - 1 / 2log33

log9(0.3)  =  - 1 / 2(1)

log9(0.3)  =  -1/2

Problem 7 :

Given log2 = 0.3010 and log3 = 0.4771, find the value of log6.

Solution :

log6  =  log(2 ⋅ 3)

log6  =  log2 + log3

Substitute the values of log2 and log3.

log6  =  0.3010 + 0.4771

log6  =  0.7781

Problem 8 :

If 2logx  =  4log3,  then find the value of 'x'.

Solution :

2logx  =  4log3

Divide each side by 2.

logx  =  (4log3) / 2

logx  =  2log3

logx  =  log32

logx  =  log9

x  =  9

Problem 9 :

If logabc  =  x, logbca  =  y and logcab  =  z, then find the value of

1/(x+1) + 1/(y+1) + 1/(z+1)

Solution :

x + 1  =  logabc + logaa  =  logaabc

y + 1  =  logbca + logbc  =  logbabc

z + 1  =  logcab + logcc  =  logcabc

1/(x+1)  =  1 / logaabc  =  logabca

1/(y+1)  =  1/logbabc  =  logabcb

1/(z+1)  =  1/logcabc  =  logabcc

1/(x+1) + 1/(y+1) + 1/(z+1)  =  logabca + logabcb + logabcc

1/(x+1) + 1/(y+1) + 1/(z+1)  =  logabcabc

1/(x+1) + 1/(y+1) + 1/(z+1)  =  1

Problem 10:

If a = log2412, b = 2436 and c = log4836, then find the value of

1 + abc

Solution :

1 + abc  =  1 + log2412  log3624 ⋅ log4836

1 + abc  =  1 + log3612 ⋅ log4836

1 + abc  =  1 + log4812

1 + abc  =  log4848 + log4812

1 + abc  =  log48(48 ⋅ 12)

1 + abc  =  log48(2 ⋅ 12)2

1 + abc  =  2log4824

1 + abc  =  2log3624 ⋅ log4836

1 + abc  =  2bc After having gone through the stuff given above, we hope that the students would have understood how to solve problems on logarithms.

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