LOGARITHM WORKSHEET

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

2. Find the value of log√264.

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

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

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

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

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

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

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

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

10. If a = log2412, b = log3624 and c = log4836, then find the value of (1 + abc) in terms of b and c.

Detailed Answer Key

1. Answer :

Write 64 as in terms of 2√2.

64 = 26

= 24+2

= 2 22

= 2⋅ [(√2)2]2

= 2⋅ (√2)4

= (2√2)4

log2√264 = log2√2(2√2)4

= 4log2√2(2√2)

= 4(1)

= 4

2. Answer :

log√264 = log√2(2)6

= 6log√2(2)

= 6log√2(√2)2

= 6 ⋅ 2log√2(√2)

= 12 ⋅ 2(1)

= 12

3. Answer :

log0.10.0001 = log0.1(0.1)4

= 4log0.10.1

= 4(1)

= 4

4. Answer :

log9(1/81) = log91 - log981

= 0 - log9(9)2

= -2log99

= -2(1)

= -2

5. Answer :

log2(0.0625) = log2(0.5)4

= 4log2(0.5)

= 4log2(1/2)

= 4(log21 - log22)

= 4(0 - 1)

= 4(-1)

= -4

6. Answer :

log9(0.3) = log9(1/3)

= log91 - log93

= 0 - log93

= -log93

= -1 / log39

= -1 / log332

= -1 / 2log33

= -1 / 2(1)

= -1/2

7. Answer :

log6 = log(2 ⋅ 3)

= log2 + log3

Substitute the values of log2 and log3.

= 0.3010 + 0.4771

= 0.7781

8. Answer :

2logx = 4log3

Divide each side by 2.

logx = 2log3

logx = log32

logx = log9

x = 9

9. Answer :

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

= logabcabc

= 1

10. Answer :

1 + abc = 1 + log2412  log3624 ⋅ log4836

= 1 + log3612 ⋅ log4836

= 1 + log4812

= log4848 + log4812

= log48(48 ⋅ 12)

= log48(2 ⋅ 12)2

= 2log4824

= 2log3624 ⋅ log4836

= 2bc

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