The solution to the logarithmic equation is - 10 / 3.
Logarithms are examples of trascendent functions, that is, function that cannot be described in algebraic terms. Herein we must take advantage of logarithm properties to solve for x in the expression given. The complete procedure is shown below:
㏒₃ (3 · x - 2) - ㏒₃ (x + 2) = 2 Given
㏒₃ [(3 · x - 2) / (x + 2)] = 2 Logarithm of a division of functions
(3 · x - 2) / (x + 2) = 3² Definitions of power and logarithm
(3 · x - 2) / (x + 2) = 9 Definition of power
3 · x - 2 = 9 · (x + 2) Definition of division / Compatibility with multiplication / Existence of the multiplicative inverse / Associative and modulative properties
3 · x - 2 = 9 · x + 18 Distributive property
- 20 = 6 · x Compatibility with addition / Existence of the additive inverse / Associative, commutative and modulative properties
6 · x = - 20 Symmetric property
x = - 20 / 6 Definition of division / Compatibility of multiplication / Existence of the multiplicative inverse / Associative, commutative and modulative properties
x = - 10 / 3 Simplification of fractions / Result
To learn more on logarithmic properties: https://brainly.com/question/12049968
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