Systems of logarithmic and exponential inequalities

Systems and unions

Each inequality gives a set on the line (or in the plane); a system is the intersection. For $\log_a$ and $a^x$ remember two modes for the base $a>1$ and $0<a<1$ — in one problem the base is the same, so the rule is the same throughout. If the statement is a union, take the union of solution sets. On exams the final answer is often written as intervals.

First the domain, then algebra; otherwise a “nice” interval may lie outside the region where $\log$ is defined

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