I'm trying to show this. Already solve a part of the exercise but in this section i don't know how to use the hint.
Let $X=(0,\infty)$, and $\lambda =$ Lebesgue measure in $\mathbb{B}_{(0,\infty)}$. Let $p \in (0, \infty)$ fixed.
Show that exist $f_p : X \rightarrow \mathbb{R}$ continuous with the following property $f_p \in L_q(\lambda)\quad (q \in (0, \infty))$ if and only if $q=p$
hint: If $p= 1$ consider the function $f_1(x) = \frac{1}{x(1+|log x|)^2}$. Consider the cases $x \in (0,1)$ and $x \in (1,\infty)$ to conclude that $f_1 \in L_1$. Use the inequality $\rho(1+log t) \leq t^\rho (t\geq 1, \rho \in (0,1))$ for $ t = \frac{1}{x}$ if $ q>1$ and $x\in(0,1)$ and for $t=x$ if $q\in (0,1)$ and $x \geq 1$