**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.

**log (m x n) = log m + log n**

**log (m / n) = log m - log n**

**log mⁿ = n log m**

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

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) ==> 4

**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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