For all [tex]\log_{x_{1}} M[/tex] there is a [tex]\log_{x_{2}}N[/tex] such that [tex]N = M^{\frac{1}{\log_{x_{2}} x_{1}} }[/tex]. The logarithmic expression log ₂ 8 is equivalent to the logarithmic expression log ₃ 27.
How to find equivalent logarithms
Logarithms are trascedental function, that is, a function that cannot be described algebraically.
Let suppose we know that [tex]\log _{x_{1}} M = \log_{x_{2}} N[/tex] such that x₁ ≠ x₂. We proceed to prove that for all [tex]\log_{x_{1}} M[/tex] there is a [tex]\log_{x_{2}} N[/tex] by logarithm properties below:
- [tex]\log_{x_{1}} M[/tex] Given
- [tex]\frac{\log_{x_{2}}M}{\log_{x_{2}}x_{1}}[/tex] Base change
- [tex]\frac{1}{\log_{x_{2}}x_{1}} \cdot \log_{x_{2}} M[/tex] Associative property
- [tex]\log_{x_{2}} M^{\frac{1}{\log_{x_{2}} x_{1}} }[/tex] [tex]n\cdot \log a = \log a^{n}[/tex]
- [tex]N = M^{\frac{1}{\log_{x_{2}} x_{1}} }[/tex] Result
For example, let suppose that x₁ = 2, M = 8 and x₂ = 3, then the value of N is:
[tex]N = 8^{\frac{1}{\log_{3} 2} }[/tex]
N = 27
Thus, for all [tex]\log_{x_{1}} M[/tex] there is a [tex]\log_{x_{2}}N[/tex] such that [tex]N = M^{\frac{1}{\log_{x_{2}} x_{1}} }[/tex]. The logarithmic expression log ₂ 8 is equivalent to the logarithmic expression log ₃ 27.
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