## EVALUATE LOGARITHMS

Evaluate logarithms :

Before going to see the example problems of evaluating logarithmic expressions, first we have to know the about the formulas used in the topic logarithms.

## Product rule in logarithms

log (m x n)  =  log m + log n

## Quotient rule in logarithms

log (m / n)  =  log m - log n

## Power property of logarithms

log m  =  n log m

## log base properties

If M, a and b are positive numbers and a ≠ 1, b  1, then

## Equality rule in logarithms

If b , x and y are positive numbers other than 1 then

if and only if x = y.

Question 1 :

Simplify log ₃ 27 + log ₃ 729

Solution :

Since the bases of both log terms are same, we can use the product rule to simplify this. That is

=  log ₃ 27 + log ₃ 729

=  log ₃ 3³ + log ₃ 3⁶

=  log ₃ (3³ x 3⁶)

=  log ₃ (3⁹)  ==>   9 log ₃ 3  ==>   9 (1)  ==>  9

Question 2 :

Simplify log ₇ 98 - log ₇ 14 + log ₇ 343

Solution :

=  log ₇ 98 - log ₇ 14 + log ₇ 343

=  log ₇ (98/14) + log ₇ 343

=  log ₇ (7) + log ₇ 7³

=  log ₇ 7 + 3 log ₇ 7

= 1 + 3 (1)  ==>

Question 3 :

Simplify (1/2) log ₉ 36 + 2 log ₉ 4 - 3 log ₉ 4

Solution :

=  (1/2) log ₉ 36 + 2 log ₉ 4 - 3 log ₉ 4

=  log ₉ √36 + log ₉ 4² - log ₉ 4³

=  log ₉ √(6 x 6) + log ₉ 16 - log ₉ 64

=  log ₉ 6 + log ₉ 16 - log ₉ 64

=  log ₉ (6 x 16) - log ₉ 64

=  log ₉ 96 - log ₉ 64

=  log ₉ (96/64)

=  log ₉ (24/16)  ==>  log ₉ (3/2)

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