Question

3 = logb(512)

`log_(b)(512)`
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Answer 1

b = 8

Explanation:

`\text{Use}\ \log_ab=c\iff a^c=b,\ \text{for}\ a>0,\ b>0\ \wedge\ a\\eq1\\\\Domain:b>0\ \wedge\ b\\eq1\\\\\log_b(512)=3\iff b^3=512\to b=\sqrt[3]{512}\\\\\begin{array}c512&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`