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The value of the logarithm log₃ (x⁻¹)[tex]^{1/b}[/tex] where x and b positive real numbers is - 5 · b⁻¹ · log₃ b.
Logarithms are trascendental functions, that is, functions that cannot be described algebraically.
In this question we must apply logarithm and algebraic properties to determine firstly the value of [tex]x[/tex] and later to find value of the latter logarithm. Now we proceed to find the value of [tex]x[/tex]:
[tex]\log_{b} x^{2} = 10[/tex]
[tex]2\cdot \log_{b} x = 10[/tex]
[tex]\log_{b} x = 5[/tex]
x = b⁵
Then, the value of the latter logarithm is:
log₃ (x⁻¹)[tex]^{1/b}[/tex]
- b⁻¹ · log₃ x
- b⁻¹ · log₃ b⁵
- 5 · b⁻¹ · log₃ b
The value of the logarithm log₃ (x⁻¹)[tex]^{1/b}[/tex] where x and b positive real numbers is - 5 · b⁻¹ · log₃ b. [tex]\blacksquare[/tex]
The statement is poorly formated, correct form is described below:
Let [tex]x[/tex] and [tex]b[/tex] be positive real numbers so that [tex]\log_{b} x^{2} = 10[/tex]. Find [tex]\log_{3} \sqrt [b]{\frac{1}{x} }[/tex].
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