Logarithm Of The Nested Radical

Algebra Level 2

log ⁡ x ( x x x x ) = ? \large \log_{\sqrt{x}} \left( \sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}} \right) = \, ?

  • Clarification: x x is a positive real number and x ≠ 1. x \neq 1.
15 2 \frac{15}2 15 4 \frac{15}{4} 15 8 \frac{15}8 15 16 \frac{15}{16}

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Abhiram Rao by Abhiram Rao , 18, India

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4 solutions

y = log ⁡ x x x x x = log ⁡ x 1 2 ( x ( x ( x ⋅ x 1 2 ) 1 2 ) 1 2 ) 1 2 = log ⁡ x 1 2 ( x ( x ( x 3 2 ) 1 2 ) 1 2 ) 1 2 = log ⁡ x 1 2 ( x ( x ⋅ x 3 4 ) 1 2 ) 1 2 = log ⁡ x 1 2 ( x ( x 7 4 ) 1 2 ) 1 2 = log ⁡ x 1 2 ( x ⋅ x 7 8 ) 1 2 = log ⁡ x 1 2 ( x 15 8 ) 1 2 = log ⁡ x 1 2 ( x 1 2 ) 15 8 = 15 8 \begin{aligned} y & = \log_{\sqrt{x}} \sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}} \\ & = \log_{x^\frac{1}{2}} \left(x \left(x \left(x \cdot{} x^\frac{1}{2} \right)^\frac{1}{2} \right)^\frac{1}{2} \right)^\frac{1}{2} \\ & = \log_{x^\frac{1}{2}} \left(x \left(x \left(x^\frac{3}{2} \right)^\frac{1}{2} \right)^\frac{1}{2} \right)^\frac{1}{2} \\ & = \log_{x^\frac{1}{2}} \left(x \left(x \cdot{} x^\frac{3}{4} \right)^\frac{1}{2} \right)^\frac{1}{2} \\ & = \log_{x^\frac{1}{2}} \left(x \left(x^\frac{7}{4} \right)^\frac{1}{2} \right)^\frac{1}{2} \\ & = \log_{x^\frac{1}{2}} \left(x \cdot{} x^\frac{7}{8} \right)^\frac{1}{2} \\ & = \log_{x^\frac{1}{2}} \left(x^\frac{15}{8} \right)^\frac{1}{2} \\ & = \log_{x^\frac{1}{2}} \left(x^\frac{1}{2} \right)^\frac{15}{8} \\ & = \boxed{\dfrac{15}{8}} \end{aligned}

55 Helpful 4 Interesting 2 Brilliant 3 Confused
Yasir Soltani
Apr 7, 2016

log ⁡ x ( x x x x ) = log ⁡ x ( x ) + log ⁡ x ( x ) + log ⁡ x ( x ) + log ⁡ x ( x ) \large \log_{\sqrt{x}} \left(\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}} \right) =\log_{\sqrt{x}}(\sqrt{x})+\log_{\sqrt{x}}\left(\sqrt{\sqrt{x}}\right)+\log_{\sqrt{x}}\left(\sqrt{\sqrt{\sqrt{x}}}\right)+\log_{\sqrt{x}}\left(\sqrt{\sqrt{\sqrt{\sqrt{x}}}}\right) = 1 + 1 2 + 1 4 + 1 8 \large =1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} = 15 8 \large =\frac{15}{8}

19 Helpful 3 Interesting 4 Brilliant 0 Confused

log ⁡ x ( x x x x ) = log ⁡ x ( x 15 16 ) = 15 16 log ⁡ x ( x ) = 15 log ⁡ x ( x ) 16 = 15 1 2 = 30 16 = 15 8 \begin{aligned}\log_{\sqrt{x}}\left(\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}\right) &= \log_{\sqrt{x}}\left(x^{\frac{15}{16}}\right) \\&= \frac{15}{16}\log_{\sqrt{x}}(x) \\&= \frac{15 \log_{\sqrt{x}}(x) }{16} \\&=\frac{15}{\frac{1}{2}} \\&= \frac{30}{16} \\&= \frac{15}{8} \end{aligned}

ADIOS!! \LARGE \text{ADIOS!!}

6 Helpful 0 Interesting 2 Brilliant 0 Confused

Hey, I might sound dumb, but how did you get x^(15/16) in the first step?

Aditya Rao - 4 years, 10 months ago

Apply the rule b 1 n = b n b^{\frac{1}{n}} = \sqrt[n]{b} , b 1 n ⋅ b = b 1 n + 1 b^{\frac{1}{n}} \cdot b=b^{\frac{1}{n}+1}

A Former Brilliant Member - 4 years, 10 months ago

In the 4th equal, fraction 15 1 2 \frac{15}{\frac{1}{2}} is incorrect. It must be 15 ⋅ 2 16 \frac{15 \cdot 2}{16} .

Alexander Israel Flores Gutiérrez - 4 years, 10 months ago
Prokash Shakkhar
Dec 31, 2016

Let, \xi= log_{\sqrt{x}} \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}} =\frac{1}{\frac{1}{2}}log_x x^{\frac{15}{16}} =\frac{15}{16}* 2 =\boxed{\color\red{\frac{15}{8}}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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