How do you find the exact value of ${log}_{2} \sqrt[4]{8}$ $log_2 root4 8$ ?

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How do you find the exact value of ${log}_{2} \sqrt[4]{8}$ $log_2 root4 8$ ?

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Answer 1 · 2017-10-01T18:42:03

$\frac{3}{4}$ $3/4$

Explanation:

$using the law of logarithms$ $"using the "color(blue)"law of logarithms"$

$∙ x {log}_{b} x = n \Leftrightarrow x = b^{n}$ $•color(white)(x)log_bx=nhArrx=b^n$

$\Rightarrow {log}_{2} (\sqrt[4]{8}) = n$ $rArrlog_2(root(4)(8))=n$

$\Rightarrow \sqrt[4]{8} = 2^{n}$ $rArrroot(4)(8)=2^n$

$using the law of exponents$ $"using the "color(blue)"law of exponents"$

$∙ x a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ $•color(white)(x)a^(m/n)=root(n)(a^m)$

$\Rightarrow \sqrt[4]{2^{3}} = 2^{\frac{3}{4}} = 2^{n}$ $rArrroot(4)(2^3)=2^(3/4)=2^n$

$since the bases on both sides are 2 we can equate$ $"since the bases on both sides are 2 we can equate"$
$the exponents$ $"the exponents"$

$\Rightarrow n = \frac{3}{4}$ $rArrn=3/4$

$\Rightarrow {log}_{2} \sqrt[4]{8} = \frac{3}{4}$ $rArrlog_2root(4)(8)=3/4$

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