Question

please help me solve the following equation

Answer 1

\[\log_x (\frac{ 1 }{ 36 })=-\frac{ 2 }{ 3 }\]

Answer 2

@Directrix

Answer 3

There are 2 ways to tackle this


Method 1: Using logs and the change of base rule


\[\Large \log_x (\frac{ 1 }{ 36 })=-\frac{ 2 }{ 3 }\]
\[\Large \frac{\log\left(\frac{1}{36}\right)}{\log(x)}=-\frac{ 2 }{ 3 }\]
\[\Large 3*\log\left(\frac{1}{36}\right) = -2*\log(x)\]
\[\Large -2*\log(x) = 3*\log\left(\frac{1}{36}\right)\]
\[\Large \log(x) = \frac{3*\log\left(\frac{1}{36}\right)}{-2}\]

I'll let you finish up


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Method 2: Change to exponential form


\[\LARGE \log_x (\frac{ 1 }{ 36 })=-\frac{ 2 }{ 3 }\]

\[\LARGE \frac{ 1 }{ 36 }=x^{{}^{-\frac{ 2 }{ 3 }}}\]

\[\LARGE \left(\frac{ 1 }{ 36 }\right)^{-\frac{3}{2}}=\left(x^{{}^{-\frac{ 2 }{ 3 }}}\right)^{-\frac{3}{2}}\]

\[\LARGE x = \left(\frac{1}{36}\right)^{-\frac{3}{2}}\]

I'll let you finish