Question

Log 6 base 20 lies in the range _________

Answer 1

log₂₀6 lies in the range ( $\frac{1}{2}$, 1).

For calculation, let us go with simple fraction's ranges.

We know that,

$log_{a} b = \frac{1}{log_{b}a } \\\\So, log_{20} 6 = \frac{1}{log_{6}20 }$

The nearest 6 power n's to the 20 are 6¹=6 and 6² = 36. As 20 lies in betweem these two numbers, we can take them as extremes.

$Their-relation-will-we, log_{6}6 < log_{20}6 < log_{36}6\\\\Using -the -log -inversion- formula -mentioned- above\\\\log_{6}6 < log_{20}6 < log_{36}6 = \frac{1}{log_{6}6} > \frac{1}{{log_{6}20}} > \frac{1}{{log_{6}36}}$

                                     $= \frac{1}{log_{6}6^{1} } > \frac{1}{{log_{6}20}} > \frac{1}{{log_{6}6^{2} }}$

                                     $= \frac{1}{1log_{6}6 } > \frac{1}{{log_{6}20}} > \frac{1}{{2log_{6}6 }} (Using,log_{a}b^{c} = clog_{a}b)\\\\= \frac{1}{1} > \frac{1}{{log_{6}20}} > \frac{1}{2} (Using,log_{a}a = 1)\\\\It-can-be-written-as: \\\\=\frac{1}{2} <\frac{1}{{log_{6}20}} < 1$

$So, log_{6}6 < log_{20}6} < log_{36}6 }} =\frac{1}{2} <\frac{1}{{log_{6}20}} < 1$

From this equation, we can finally say that

log₂₀6 lies in between ( $\frac{1}{2}$, 1).

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