If $$ \log_x (0.1) = -\frac{1}{3} $$, find the value of $$ x $$.

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Accepted Answer

We are given $$ \log_x (0.1) = -\frac{1}{3} $$. Using the logarithmic definition, this can be rewritten as:  
$$
x^{-\frac{1}{3}} = 0.1
$$

Taking the reciprocal of both sides:  
$$
x^{\frac{1}{3}} = \frac{1}{0.1}
$$

Since $$ \frac{1}{0.1} = 10 $$:  
$$
x^{\frac{1}{3}} = 10
$$

Now, cube both sides to isolate $$ x $$:  
$$
x = 10^3 = 1000
$$

Thus, the value of $$ x $$ is **1000**.

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