Lesson 8 of 9, 10 min read
Logarithmic inequalities
Solving logarithmic inequalities by combining the domain with an inequality between the arguments, keeping or reversing the sign depending on the base.
Two conditions at once
A logarithmic inequality always has two parts:
- the domain (every argument positive);
- the inequality between the arguments, with its direction set by the base.
The solution is where both hold.
Exponential inequalities with any base
Logs also handle exponential inequalities like \(3^x < 10\). Taking \(\log_3\) (base greater than \(1\), so the sign stays): \(x < \log_3 10 \approx 2.096\).
Practice
Quick check
What is the solution of \(\log x < 0\)?
\(\log x < \log 1\) gives \(x < 1\), and the domain requires \(x > 0\).
Quick check
To solve \(\log_{0.2}(x) < \log_{0.2}(4)\), you should solve:
With a base between \(0\) and \(1\) the sign reverses.
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