The logarithmic function is ㏒ 12³ = ㏒ x⁴ · y³ · z⁶. The equation into the expanded logarithmic form is 3 · ㏒ 12 = 4 · ㏒ x + 3 · ㏒ y + 6 · ㏒ z.
How to apply logarithms on a exponential function
Logarithms are the reciprocal of exponential functions. In this question we must apply logarithms and its properties to obtain a linear-like expression in the following lines. The logarithm properties are summarized below:
㏒ₐ (b/c) = ㏒ₐ b - ㏒ₐ c (1)
㏒ₐ (b · c) = ㏒ₐ b + ㏒ₐ c (2)
㏒ₐ bⁿ = n · ㏒ₐ b (3)
a) To obtain the corresponding logarithmic function we apply logarithms on both sides of the expression:
㏒ 12³ = ㏒ x⁴ · y³ · z⁶ (4)
The logarithmic function is ㏒ 12³ = ㏒ x⁴ · y³ · z⁶. [tex]\blacksquare[/tex]
b) The expanded logaritmic form of (4) consists in the addition and subtraction of logarithms of non-power numbers, whose formed could be found, if possible, by logarithm properties:
㏒ 12³ = ㏒ x⁴ · y³ · z⁶
㏒ 12³ = ㏒ x⁴ + ㏒ y³ + ㏒ z⁶
3 · ㏒ 12 = 4 · ㏒ x + 3 · ㏒ y + 6 · ㏒ z
The equation into the expanded logarithmic form is 3 · ㏒ 12 = 4 · ㏒ x + 3 · ㏒ y + 6 · ㏒ z. [tex]\blacksquare[/tex]
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