## EVALUATING LOGARITHMIC EXPRESSIONS

Evaluating Logarithmic Expressions :

In this section, you will learn how to evaluate logarithmic expressions.

To evaluate logarithmic expressions, you must be aware of the following three fundamentals laws of logarithms.

Law 1 :

Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base.

That is,

logamn  =  logam + logan

Law 2 :

Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base.

That is,

loga(m/n)  =  logam - logan

Law 3 :

Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.

That is,

logamn  =  nlogam

Change of Base :

logba  =  logx⋅ logbx

logba  =  logxa / logxb

## Evaluating Logarithmic Expressions - Examples

Example 1 :

Evaluate :

log327 + log3729

Solution :

=  log327 + log3729

=  log333 + log336

=  3log33 + 6log33

=  3(1) + 6(1)

=  3 + 6

=  9

Example 2 :

Simplify :

log108 + log105 - log104

Solution :

log108 + log105 - log104

=  log10 (8 ⋅ 5)/4]

log1010

=  1

Example 3 :

Evaluate :

log798 - log714 + log7343

Solution :

=  log798 - log714 + log7343

=  log7(98/14) + log7343

=  log77 + log773

=  1 + 3log77

=  1 + 3(1)

=  1 + 3

=  4

Example 4 :

Evaluate :

(1/2)log936 + 2log94 - 3log94

Solution :

=  (1/2)log936 + 2log94 - 3log94

=  log9(36)1/2 + log942 - log943

=  log9√36 + log916 - log964

=  log96 + log916 - log964

=  log9(6 ⋅ 16) - log964

=  log996 - log964

=  log9(96/64)

=  log9(3/2)

Example 5 :

Simplify :

log3 log4⋅ log5⋅ log6⋅ log7⋅ log87

Solution :

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

Example 6 :

Simplify :

log721 + log777 + log788 - log7121 - log724

Solution :

=  log721 + log777 + log788 - log7121 - log724

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

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

=  log142296 - log72904

=  log7(142296 / 2904)

=  log749

=  log772

=  2log77

=  2(1)

=  2

Example 7 :

Simplify :

log816 + log852 - 1/log138

Solution :

log816 + log852 - 1/log138

=  log816 + log852 - log813

=  log8[(16 ⋅ 52)/13]

=  log8(16 ⋅ 4)

=  log864

=  log882

=  2log88

=  2(1)

=  2

Example 8 :

Simplify :

5log102 + 2log103 - 6log644

Solution :

5log102 + 2log103 - (2⋅3) log644

log1025 + log1032 - 2log6443

log1032 + log109 - 2log6464

=  log1032 + log109 - 2(1)

log1032 + log109 - 2log1010

=  log10(32  9) - log10102

log10288 - log10100

log10(288/100)

=  log10(72/25)

After having gone through the stuff given above, we hope that the students would have understood how to evaluate logarithmic expressions

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