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»INEGALITATI

Data publicarii: 17 Iulie, 2010

Suport teoretic:

Logaritmi, radicali, puteri rationale.

Enunt:

Sa se demonstreze inegalitatea:

${{log}_{15}{25}}<{\sqrt[5]{15}}.$ {{log}_{15}{25}}<{\sqrt[5]{15}}.

Demonstratie:

Observam ca:

$1={{log}_{15}{15}}<{{log}_{15}{25}}<{{log}_{15}{{15}^2}}=2,\;(1)\;si$ 1={{log}_{15}{15}}<{{log}_{15}{25}}<{{log}_{15}{{15}^2}}=2,\;(1)\;si

$1={\sqrt[5]{1}}<{\sqrt[5]{15}}<{\sqrt[5]{2^5}}=2.\;(2)$ 1={\sqrt[5]{1}}<{\sqrt[5]{15}}<{\sqrt[5]{2^5}}=2.\;(2)

Sa identificam un numar rational r din intervalul (1;2), astfel incat:

${1}<{\sqrt[5]{15}}<{r}<{{log}_{15}{25}}<{2}.\;(3)$ {1}<{\sqrt[5]{15}}<{r}<{{log}_{15}{25}}<{2}.\;(3)

Prin tatonari, se poate alege r=4/3 si deducem:

${{\sqrt[5]{15}}>{\frac{4}{3}}}{\Leftrightarrow}{{15}>{(\frac{4}{3})}^5}{\Leftrightarrow}{\cdots}{\Leftrightarrow}{{3645}>{1024}},\;adevarat;(4)$ {{\sqrt[5]{15}}>{\frac{4}{3}}}{\Leftrightarrow}{{15}>{(\frac{4}{3})}^5}{\Leftrightarrow}{\cdots}{\Leftrightarrow}{{3645}>{1024}},\;adevarat;(4)
${{{log}_{15}{25}}<{\frac{4}{3}}}{\Leftrightarrow}{{log}_{15}{25}}<{{log}_{15}{15}}^{\frac{4}{3}}{\Leftrightarrow}{{25}<{15}^{\frac{4}{3}}}{\Leftrightarrow}{\cdots}{\Leftrightarrow}{{15625}<{50625}},\;adevarat;(5).$ {{{log}_{15}{25}}<{\frac{4}{3}}}{\Leftrightarrow}{{log}_{15}{25}}<{{log}_{15}{15}}^{\frac{4}{3}}{\Leftrightarrow}{{25}<{15}^{\frac{4}{3}}}{\Leftrightarrow}{\cdots}{\Leftrightarrow}{{15625}<{50625}},\;adevarat;(5).