Hvordan løse eksponentialulikheter

Ved eksponentialulikheter er det svært viktig at du ser på verdien til argumentet $a$ til logaritmen. Den verdien avgjør om du må snu ulikheten når du ganger eller deler med $\mathrm{ln}a$ eller $\mathrm{lg}a$.

Regel

Slikløserdueksponentialulikheter

Når $a>1$ er $\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(a\right)>0$ og du kan løse ulikheten på vanlig måte.

$\begin{array}{llllllll}\hfill {a}^{x}& >b\phantom{\rule{2em}{0ex}}& \hfill {a}^{x}& >b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mathrm{ln}{a}^{x}& >\mathrm{ln}b\phantom{\rule{2em}{0ex}}& \hfill \mathrm{lg}{a}^{x}& >\mathrm{lg}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\mathrm{ln}a& >\mathrm{ln}b\phantom{\rule{2em}{0ex}}& \hfill x\mathrm{lg}a& >\mathrm{lg}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& >\frac{\mathrm{ln}b}{\mathrm{ln}a}\phantom{\rule{2em}{0ex}}& \hfill x& >\frac{\mathrm{lg}b}{\mathrm{lg}a}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Når $0 er $\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(a\right)<0$ og da du snu ulikheten, siden du ender opp med å gange eller dele med et negativt tall!

$\begin{array}{llllllll}\hfill {a}^{x}& >b\phantom{\rule{2em}{0ex}}& \hfill {a}^{x}& >b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mathrm{ln}{a}^{x}& >\mathrm{ln}b\phantom{\rule{2em}{0ex}}& \hfill \mathrm{lg}{a}^{x}& >\mathrm{lg}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\mathrm{ln}a& >\mathrm{ln}b\phantom{\rule{2em}{0ex}}& \hfill x\mathrm{lg}a& >\mathrm{lg}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& <\frac{\mathrm{ln}b}{\mathrm{ln}a}\phantom{\rule{2em}{0ex}}& \hfill x& <\frac{\mathrm{lg}b}{\mathrm{lg}a}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Eksempel 1

Løs ulikheten ${\text{}3,5\text{}}^{x}>439$

$\begin{array}{llllll}\hfill {\text{}3,5\text{}}^{x}& >439\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mathrm{ln}{\text{}3,5\text{}}^{x}& >\mathrm{ln}439\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\mathrm{ln}\text{}3,5\text{}& >\mathrm{ln}439\phantom{\rule{2em}{0ex}}& \hfill & |:\mathrm{ln}\text{}3,5\text{}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& >\frac{\mathrm{ln}439}{\mathrm{ln}\text{}3,5\text{}}\approx \text{}4,9\text{}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Eksempel 2

Løs ulikheten $50\cdot {\text{}1,05\text{}}^{x}>300$

$\begin{array}{llllllll}\hfill 50\cdot {\text{}1,05\text{}}^{x}& >300\phantom{\rule{2em}{0ex}}& \hfill & |:50\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\text{}1,05\text{}}^{x}& >6\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mathrm{ln}{\text{}1,05\text{}}^{x}& >\mathrm{ln}6\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\mathrm{ln}\text{}1,05\text{}& >\mathrm{ln}6\phantom{\rule{2em}{0ex}}& \hfill & |:\mathrm{ln}\text{}1,05\text{}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& >\frac{\mathrm{ln}6}{\mathrm{ln}\text{}1,05\text{}}\approx \text{}36,7\text{}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Eksempel 3

Løs ulikheten $3\cdot {\text{}0,25\text{}}^{x}>27$

$\begin{array}{llllllll}\hfill 3\cdot {\text{}0,25\text{}}^{x}& >27\phantom{\rule{2em}{0ex}}& \hfill & |:3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\text{}0,25\text{}}^{x}& >9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \mathrm{lg}{\text{}0,25\text{}}^{x}& >\mathrm{lg}9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\mathrm{lg}\text{}0,25\text{}& >\mathrm{lg}9\phantom{\rule{2em}{0ex}}& \hfill & |:\mathrm{lg}\text{}0,25\text{}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& <\frac{\mathrm{lg}9}{\mathrm{lg}\text{}0,25\text{}}\approx \text{}-1,6\text{}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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