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Simplification guidée avec deux bases

#logarithme#base-a#simplification

\log_2

Exercice

On simplifie :

\log_3\left(\dfrac{15}{4}\right) + \log_2\left(\dfrac{1}{27}\right) + \log_3\left(\dfrac{4}{5}\right)

Solution.

= \log_3\left(\dfrac{15}{4}\times\dfrac{4}{5}\right) + \log_2(3^{-3})

= \log_3(3) - \log_2(3^3)

= 1 - \log_2(27)

= \log_2(2) - \log_2(27)

= \log_2\left(\dfrac{2}{27}\right)

Exercice

On simplifie :

\log_3\left(\dfrac{15}{4}\right) + \log_2\left(\dfrac{1}{27}\right) + \log_3\left(\dfrac{4}{5}\right)

Solution.

= \log_3\left(\dfrac{15}{4}\times\dfrac{4}{5}\right) + \log_2(3^{-3})

= \log_3(3) - \log_2(3^3)

= 1 - \log_2(27)

= \log_2(2) - \log_2(27)

= \log_2\left(\dfrac{2}{27}\right)