8. Exponential And Logatithmic Inequality Solver — Theory
Exponential and logarithmic functions are monotone (increasing or decreasing), depending on the base and on how \(x\) appears.
This decides how inequalities behave when we apply \(\log\) or exponentiate.
1) Monotonicity rules
- \(a^x\) is increasing if \(a>1\), decreasing if \(0
- \(\log_a(x)\) is increasing if \(a>1\), decreasing if \(0
- \(a^{k x+\varphi}\) is increasing if \(k\ln(a)>0\) and decreasing if \(k\ln(a)<0\).
2) Domain (logs)
For \(\log_b(kx+\varphi)\), we must have:
\[
kx+\varphi > 0.
\]
The calculator draws red dashed vertical lines where \(kx+\varphi=0\) to mark domain boundaries.
3) What the calculator does
- Builds \(f(x)=\mathrm{LHS}(x)-\mathrm{RHS}(x)\).
- Splits the domain at log boundaries and intersection points \(f(x)=0\).
- Checks the sign of \(f\) on each sub-interval to keep where \(f(x)\ \square\ 0\).
- Plots both sides, plus the solution as thick green segments on the \(x\)-axis and on a number line.
4) Worked example (same as “Fill example”)
\[
2^{x+1}-5 \le 3^x
\qquad\Longleftrightarrow\qquad
f(x)=2^{x+1}-5-3^x \le 0.
\]
The tool finds approximate intersections in the chosen domain and keeps the intervals where the inequality is satisfied.
