Question

3 = logb(512)_{b}(512) " align="absmiddle" class="latex-formula">
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Answer 1

Answer:

b = 8

Step-by-step explanation:

$\text{Use}\ \log_ab=c\iff a^c=b,\ \text{for}\ a>0,\ b>0\ \wedge\ a\neq1\\\\Domain:b>0\ \wedge\ b\neq1\\\\\log_b(512)=3\iff b^3=512\to b=\sqrt[3]{512}\\\\\begin{array}{c|c}512&2\\256&2\\128&2\\64&2\\32&2\\16&2\\8&2\\4&2\\2&2\\1\end{array}\\\\512=2^9=2^{3\cdot3}=(2^3)^3=8^3\\\\\text{Therefore}\ b=\sqrt[3]{512}=\sqrt[3]{8^3}=8\in D$

$\text{Used:}\\\\(a^n)^m=a^{nm}\\\\\sqrt[n]{a^n}=a$