Rule for the direction in inequalities

When one “removes” the logarithms from an inequality, the direction of the inequality between the arguments depends on the base.

Property — Direction and base

$\log_a f(x) \lessgtr \log_a g(x)$ is equivalent, within the ECs ($f, g > 0$), to:

• $f(x) \lessgtr g(x)$ if $a > 1$ (same direction);
• $f(x) \gtrless g(x)$ if $0 < a < 1$ (direction reversed).

Example — $\log_{1/2}(x-3) > 1$

EC: $x - 3 > 0 \iff x > 3$.

Uniform writing: $1 = \log_{1/2}(1/2)$. Hence $\log_{1/2}(x-3) > \log_{1/2}(1/2)$.

Base $<1$, direction reversed: $x - 3 < \tfrac{1}{2} \iff x < \tfrac{7}{2}$.

Intersection with the EC: $3 < x < \tfrac{7}{2}$, that is $\boxed{x\in \left(3;\tfrac{7}{2}\right)}$.

Links

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