(\log_b \sqrt[3]{a})(\log_c b^4)(\log_a \sqrt[4]{c^3}) = 1
Answer (1)
To solve the equation:
( lo g b 3 a ) ( lo g c b 4 ) ( lo g a 4 c 3 ) = 1
we can start by simplifying each logarithmic term individually.
Simplify each logarithm using basic properties:
The first term is lo g b 3 a . We can use the property of logarithms that states lo g b a 1/ n = n 1 lo g b a : lo g b 3 a = 3 1 lo g b a
The second term is lo g c b 4 . Applying the power rule for logarithms, lo g c b n = n lo g c b : lo g c b 4 = 4 lo g c b
The third term is lo g a 4 c 3 . Similarly, use the property lo g a c 3/4 = 4 3 lo g a c : lo g a 4 c 3 = 4 3 lo g a c
Substitute back into the original equation:
The equation now becomes: ( 3 1 lo g b a ) ( 4 lo g c b ) ( 4 3 lo g a c ) = 1
Simplifying this product: 3 1 ⋅ 4 ⋅ 4 3 ⋅ lo g b a ⋅ lo g c b ⋅ lo g a c = 1
Here, multiply the coefficients: 1 ⋅ lo g b a ⋅ lo g c b ⋅ lo g a c = 1
Apply the change of base property:
Using the identity lo g b a ⋅ lo g c b ⋅ lo g a c = 1 , which is derived from the cylindrical symmetry of logarithms, the equation holds. This is a well-known cyclical property of logarithms.
Thus, this problem illustrates the use of properties of logarithms and simplification to confirm the identity of this mathematical expression.
