$\log_{0.1}(x^4) - 4 \geq 0$ - Solution verification

Question

My question: are these steps ok?
To be more precise, is the step where I take fourth root ok? $$\log_{0.1}(x^4) - 4 \geq 0$$ $$\iff$$ $$\log_{0.1}(x^4) \geq 4$$ $$\iff$$ $$\log_{0.1}(x^4) \geq \log_{0.1}\left(\frac{1}{10000}\right)$$ $$\iff$$ $$x^4 \leq \frac{1}{10000}$$ $$\iff$$ $$|x|\leq \frac{1}{10}$$ This is the solution without $x=0$.

Answer 1

It is correct, but it would be easier to take into account that $\log_{0.1}(x^4)=4\log_{0.1}(x)$.

Answer 2

It’s completely fine, and you also correctly used the absolute value when taking the fourth root, but you can solve the inequality more quickly by noting:

$$\log_a b^c = c\log_a \vert b\vert \tag{1}$$

$$\log_{a^b} c = \frac{1}{b}\log_a c \tag{2}$$

So, using $0.1 = 10^{-1}$, you can rewrite $\log_{0.1} \left(x^4\right) \geq 4$ as follows:

$$\log_{0.1} \left(x^4\right) \geq 4 \iff -4\log_{10} \vert x\vert \geq 4 \iff \log_{10} \vert x\vert \leq -1 \iff \vert x\vert \leq \frac{1}{10}$$

Of course, as mentioned, your way is completely fine as well, but perhaps this way is a bit faster.

Answer 3

For fun, an option :

$z:= \log_{0.1}(x^4) \ge 4;$

$x^4= (0.1)^z=(10^{-1})^z= (10)^{-z}$, i.e.

$(10)^z=x^{-4} \ge (10)^4,$

finally $x \le 1/(10)$.

Your answer is fine as has been pointed out.