Simplify/Condense log base 3 of 108+ log base 3 of 3/4

Problem

(log_3)(108)+(log_3)(3/4)

Solution

1. Identify the logarithmic property for addition, which states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments.

(log_b)(M)+(log_b)(N)=(log_b)(M⋅N)

2. Apply the product property to combine the two terms into a single logarithm.

(log_3)(108⋅3/4)

3. Multiply the arguments inside the logarithm.

108⋅3/4=27⋅3

27⋅3=81

4. Rewrite the expression with the simplified argument.

(log_3)(81)

5. Evaluate the logarithm by expressing 81 as a power of the base 3.

81=3

(log_3)(3)=4

Final Answer

(log_3)(108)+(log_3)(3/4)=4

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