Basic cases with base less than 1

For the logarithm too, a base less than $1$ implies a reversed direction (and the argument must remain $>0$). The five examples that follow cover the standard cases.

Example — $\log_{1/3}(x+1)\le 2$

EC: $x+1>0\iff x>-1$. I rewrite $2=\log_{1/3}(1/9)$: $\log_{1/3}(x+1)\le\log_{1/3}(1/9).$ Base $<1$, direction reversed: $x+1\ge 1/9 \iff x\ge -8/9$. Intersection with the EC: $\boxed{x\ge -8/9}$.

Example — Decimal base $1/10$

$\log_{1/10}(2x-3)>-1$. EC: $2x-3>0\iff x>3/2$. With $-1=\log_{1/10}(10)$: $\log_{1/10}(2x-3)>\log_{1/10}(10) \iff 2x-3<10 \iff x<13/2.$ Intersection: $\boxed{3/2<x<13/2}$.

Example — Substitution with base $<1$

$(\log_{1/2}x)^2 - 3\log_{1/2}x + 2 \ge 0$. I set $t=\log_{1/2}x$ (with EC $x>0$): $t^2-3t+2\ge 0 \iff t\le 1 \vee t\ge 2.$ I go back to $x$:

• $\log_{1/2}x\le 1 \iff x\ge 1/2$ (direction reversed, base $<1$);
• $\log_{1/2}x\ge 2 \iff x\le 1/4$ (direction reversed).

Together with the EC ($x>0$): $\boxed{0<x\le 1/4 \vee x\ge 1/2}$.

Example — Comparison of two logs with base $<1$

$\log_{1/4}(x-2)>\log_{1/4}(3x-7)$. EC: $x-2>0$ and $3x-7>0\iff x>7/3$. Base $<1$, direction reversed: $x-2<3x-7 \iff 5<2x \iff x>5/2$. Intersection with the EC: $\boxed{x>5/2}$.

Example — Comparison with $0$

$\log_{0,5}(x^2-4)>0$. EC: $x^2-4>0\iff x<-2\vee x>2$. With $0=\log_{0,5}1$, base $<1$, direction reversed: $x^2-4<1 \iff x^2<5\iff -\sqrt{5}<x<\sqrt{5}$. Intersection with the EC: $\boxed{-\sqrt{5}<x<-2 \vee 2<x<\sqrt{5}}$.

Links

Topics: Logarithmic function
Concepts: Existence conditions · Logarithmic inequalities · Logarithm
Functions: Logarithmic function
Skills: Reason by cases · Solve inequalities