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Question:
The absolute value of [tex]\sf{log_{\frac{1}{2}}\:log_{\sqrt{5}}\:log_{3}(243)}[/tex] is equal to
[tex]\\[/tex]
Answer:
-1
[tex]\\[/tex]
Step-by-step explanation:
First, let's solve the innermost part of this question.
log₃(243)
⇒ log₃(3⁵)
⇒ 5 log₃(3)
⇒ 5 × 1
⇒ 5
[tex]\\[/tex]
Now, the equation we have is:
[tex]\sf{log_{\frac{1}{2}}\:log_{\sqrt{5}}(5)}\\\\[/tex]
Now,
[tex]\sf{log_{\sqrt{5}}(5)}\\\\[/tex]
[tex]= \sf{log_{\sqrt{5}}({\sqrt{5}})^{2}}\\\\[/tex]
[tex]= \sf{2log_{\sqrt{5}}(\sqrt{5})}\\\\[/tex]
= 2 × 1
= 2
[tex]\\[/tex]
For solving the outermost part:
[tex]\sf{Let\:log_{\frac{1}{2}}(2)=x}\\\\[/tex]
[tex]\implies \sf{2=\bigg(\dfrac{1}{2}\bigg)^{x}}\\\\[/tex]
[tex]\implies \sf{2^{1}=2^{-x}}\\\\[/tex]
[tex]\implies \sf{-x = 1}\\\\[/tex]
[tex]\therefore \boxed{\bf{x=-1}}\\\\[/tex]