SIMPLIFYING LOGARITHMIC EXPRESSIONS WORKSHEET

Simplify each of the following expressions :

Problem 1 :

log5 25 +  log5 625

Problem 2 :

log5 4 +  log5 (1/100)

Problem 3 :

log8 128 -  log16

Problem 4 :

log3 2  log4 3 ⋅ log5 4 ⋅ log6 5 ⋅ log7 6 ⋅ log8 7

Problem 5 :

log7 21 + log7 77 + log7 88 - log7 121 - log7 24

Problem 6 :

log8 16 + log8 52 - 1/log13 8

Problem 7 :

5log10 2 + 2log10 3 - 6log64 4

Problem 8 :

log10 8 + log10 5 - log10 4

Answers

1. Answer :

=  log525 + log5625 

=  log5(25 ⋅ 625)

=  log5(52 ⋅ 54)

=  log55(2 + 4)

=  log556

 =  6log55

=  6(1)

=  6

2. Answer :

=  log54 + log5(1/100) 

=  log5(4 ⋅ 1/100)

=  log5(1/25)

=  log5(1/52)

=  log55-2

=  -2log55

=  -2(1)

=  -2

3. Answer :

=  log8128 -  log816

=  log8(128/16)

=  log88

=  1

4. Answer :

log3 log4⋅ log5⋅ log6⋅ log7⋅ log87

In the given expression, logarithms have bases.

First group the logarithms with the same base and simplify.

= (log3 log43) ⋅ (log5⋅ log65) ⋅ (log7⋅ log87)

= log4⋅ log64  log86

= log6 log86

= log82

= 1/log28

= 1/log223

= 1/3(log22)

= 1/3(1)

= 1/3

5. Answer :

=  log721 + log777 + log788 - log7121 - log724

=  log7(21 ⋅ 77 ⋅ 88) - (log7121 + log724)

 =  log7(21 ⋅ 77 ⋅ 88) - log7(121 ⋅ 24)

=  log7142296 - log72904

=  log7(142296/2904)

=  log749

=  log772

=  2log77

=  2(1)

=  2

6. Answer :

=  log816 + log852 - 1/log138

 log816 + log852 - log813

=  log8[(16 ⋅ 52)/13]

=  log8(16 ⋅ 4)

=  log864

=  log882

=  2log88

=  2(1)

=  2

7. Answer :

=  5log102 + 2log103 - 6log644

=  5log102 + 2log103 - (2 ⋅ 3) log644

 log10 25log10 32 - 2log64 43

 log1032 + log109 - 2log6464

 log1032 + log109 - 2 (1)

 log1032 + log109 - 2log1010

=  log10(32  9) - log10102

=  log10(288/100)

=  log10(72/25)

8. Answer :

=  log108 + log105 - log104

=  log10[(8 ⋅ 5)/4]

=  log10(40/4)

 log1010

=  1

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