How do I solve this logarithm problem?

Question

I'm trying to solve this problem:

If $\log_{27}(a)=b$, find $\log_{\sqrt[6]{a}}\sqrt{3}$

However, I'm unable to see any connection in those given information. How can I solve this logarithm?

Answer 1

Given $\log_{27}(a) = b$ then $a = 27^b$. Then: $$\sqrt[6]{a} = (27^b)^\frac{1}{6} = (3^{3 \cdot \frac{1}{6}})^b = 3^\frac{b}{2}$$ $$\log_{\sqrt[6]{a}}x = \log_{3^\frac{b}{2}}x = \frac{\log_{3}x}{\log_{3}3^\frac{b}{2}} = \frac{2}{b}\log_{3}x$$ Since $x = \sqrt{3}$ then $$\frac{2}{b}\log_{3}x = \frac{2}{b}\cdot\frac{1}{2} = \frac{1}{b}$$