# LOGARITHM PROBLEMS

Problem 1 :

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

Solution :

Write 64 as in terms of 2√2.

64  =  26

=  24+2

=  2 22

=  2⋅ [(√2)2]2

=  2⋅ (√2)4

=  (2√2)4

Then,

log2√264  =  log2√2(2√2)4

=  4log2√2(2√2)

=  4(1)

=  4

Problem 2 :

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

=  logabcabc

=  1

Problem 3 :

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

Solution :

1 + abc  =  1 + log2412  log3624 ⋅ log4836

=  1 + log3612 ⋅ log4836

=  1 + log4812

=  log4848 + log4812

=  log48(48 ⋅ 12)

=  log48(2 ⋅ 12)2

=  2log4824

=  2log3624 ⋅ log4836

=  2bc

Problem 4 :

Find the value of log 0.0001 to the base 0.1.

Solution :

log0.10.0001  =  log0.1(0.1)4

=  4log0.10.1

=  4(1)

=  4

Problem 5 :

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 6 :

Find the value of log√264.

Solution :

log√264  =  log√2(2)6

=  6log√2(2)

=  6log√2(√2)2

=  6 ⋅ 2log√2(√2)

=  12 ⋅ 2(1)

=  12

Problem 7 :

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

Solution :

log9(1/81)  =  log91 - log981

=  0 - log9(9)2

=  -2log99

=  -2(1)

=  -2

Problem 8 :

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

Solution :

log2(0.0625)  =  log2(0.5)4

=  4log2(0.5)

=  4log2(1/2)

=  4(log21 - log22)

=  4(0 - 1)

=  4(-1)

=  -4

Problem 9 :

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

Solution :

log6  =  log(2 ⋅ 3)

=  log2 + log3

Substitute the values of log2 and log3.

=  0.3010 + 0.4771

=  0.7781

Problem 10 :

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

Solution :

log9(0.3)  =  log9(1/3)

=  log91 - log93

=  0 - log93

=  -log93

=  -1 / log39

=  -1 / log332

=  -1 / 2log33

=  -1 / 2(1)

=  -1/2 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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