# SIMPLIFYING LOGARITHMIC EQUATIONS WORKSHEET

Simplifying logarithmic equations :

Here we are going to see how to simplify logarithmic equations.

Before look into the example problems, we have to know about the formulas in logarithms.

Product rule :

log a (m ⋅ n)  =  logam + logan

Quotient rule :

log a (m / n)  =  logam - logan

Power rule :

log a mn  =  n logam

Change of base rule :

log a m  =  logb⋅ logab

log a a  =  1

Based on the rules given above, we have to simplify the logarithmic equations.

## Simplifying logarithmic equations - Examples

Example 1 :

Simplify log5 25 +  log5 625

Solution :

=  log5 25 +  log5 625

=  log5 (25 ⋅ 625)

=  log5 (52 ⋅ 54)

=  log5 5(2 + 4)

=  log5 56

=  6log5 5

=  6 (1)

=  6

Example 2 :

Simplify log5 4 +  log5 (1/100)

Solution :

=  log5 4 +  log5 (1/100)

=  log5 (4 ⋅ (1/100))

=  log5 (1/25)

=  log5 5-2

=  -2log5 5

=  -2(1)

=  -2

Example 3 :

Simplify log8 128 -  log16

Solution :

=  log8 128 -  log16

=  log8 (128/16)

=  log8 8

=  1

Example 4 :

Simplify log3 2  log4⋅ log5⋅ log6⋅ log7⋅ log8 7

Solution :

The given expression is having different bases, to combine them let us group them as pairs.

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

=  log4 2 ⋅ log6 4  log8 6

=  log6 2  log8 6

=  log8 2

Using change base rule, we are going to interchange the base.

=  1/log2 8

=  1/log2 23

=  1/3(log2 2)

=  1/3(1)

=  1/3

Example 5 :

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

Solution :

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

Since the given expression has same base, we may group them easily.

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

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

=  log7 [(21 ⋅ 77 ⋅ 88)/(121 ⋅ 24)]

=  log7 49

=  log7 72

=  2 log7 7

=  2 (1)  =  2

Example 6 :

Simplify log8 16 + log8 52 - 1/log13 8

Solution :

=   log8 16 + log8 52 - 1/log13 8

Let us use change base rule, we are going to interchange the base.

=   log8 16 + log8 52 - log8 13

=   log8 [(16 ⋅ 52)/13]

=   log8 (16 ⋅ 4)

=   log8 64

=   log8 82

=   2log8 8

=  2(1)

=  2

Example 7 :

Simplify 5 log10 2 + 2 log10 3 - 6log64 4

Solution :

=   5 log10 2 + 2 log10 3 - (2⋅3) log64 4

=   log10 25log10 32 - 2log64 43

=   log10 32 + log10 9 - 2log64 64

=   log10 32 + log10 9 - 2 (1)

=   log10 32 + log10 9 - 2 log1010

=   log10 (32  9) - log10102

=   log10 (288/100)

=   log10 (72/25)

Example 8 :

Simplify  log10 8 + log10 5 - log10 4

Solution :

=   log10 8 + log10 5 - log10 4

=   log10 [(8 ⋅ 5)/4]

=   log10 10

=  1

After having gone through the stuff given above, we hope that the students would have understood "Simplifying logarithmic equations".

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