Applicando le propriet dei logaritmi, risolvere la seguente espressione:
[math] frac(\\log_(\sqrt2) 5 - \\log_2 {25})(\\log_4 (5)) + frac(\\log_3(2) + 4\\log_9(4))(\\log_(27) (8)) [/math]
Svolgimento
Sappiamo che la radice di un numero pu essere scritta in questo modo:
[math] \sqrt{a} = a^{1/2}[/math]
quindi
[math] frac(\\log_(2^{1/2}) 5 - \\log_2 (25))(\\log_(2^2) (5)) + frac(\\log_3(2) + 4\\log_(3^2)(4))(\\log_(3^3) (8)) [/math]
Secondo un propriet dei logaritmi, sappiamo che:
[math] \\log_a (b) = frac(\\log_c (b))(\\log_c (a)) [/math]
Di conseguenza, abbiamo che:
[math] frac(frac(\\log_2 (5))(\\log_2 (\sqrt2)) - \\log_2 {25})(frac(\\log_2(5))(\\log_2 (4))) + frac(\\log_3(2) + 4\\log_(9)(4))(frac(\\log_3(8))(\\log_3(3^3))) [/math]
Secondo le propriet dei logaritmi, abbiamo che :
[math] \\log_a(b^k) = k \\log_a (b) [/math]
Possiamo quindi mettere ad esponente il coefficiente del logaritmo:
[math] frac(frac(\\log_2 (5))(\\log_2 (\sqrt2)) - \\log_2 {25})(frac(\\log_2(5))(\\log_2 (4))) + frac(\\log_3(2) + \\log_(9)(4^4))(frac(\\log_3(8))(\\log_3(3^3))) [/math]
[math] frac(frac(\\log_2 (5))(\\log_2 (\sqrt2)) - \\log_2 {25})(frac(\\log_2(5))(\\log_2 (4))) + frac(\\log_3(2) + frac(\\log_3(4^4))(\\log_3(3^2)))(frac(\\log_3(8))(\\log_3(3^3))) [/math]
Possiamo calcolare il valore di alcuni logaritmi:
[math] \\log_2 (\sqrt2) = \\log_2 {2^{1/2}} = 1/2 [/math]
[math] \\log_2 (4) = \\log_2 (2^2) = 2 [/math]
[math] \\log_3 (27) = \\log_3 (3^3) = 3 [/math]
[math] \\log_3 (9) = \\log_3 (3^2) = 2 [/math]
Sostituiamo:
[math] frac(frac(\\log_2 (5))(1/2) - \\log_2 (25))(frac(\\log_2(5))(\\log_2 (4))) + frac(\\log_3(2) + frac(\\log_3(4^4))(2))(frac(\\log_3(8))(3)) [/math]
[math] frac( 2\\log_2 (5) - \\log_2 (25))(frac(\\log_2(5))(2)) + frac(\\log_3(2) + frac(\\log_3(4^4))(2))(frac(\\log_3(8))(3)) [/math]
[math] frac( \\log_2 (5^2) - \\log_2 (25))(frac(\\log_2(5))(2)) + frac(\\log_3(2) + frac(\\log_3(4^4))(2))(frac(\\log_3(8))(3)) [/math]
[math] frac( \\log_2 (25) - \\log_2 (25))(frac(\\log_2(5))(2)) + frac(\\log_3(2) + frac(\\log_3(4^4))(2))(frac(\\log_3(8))(3)) [/math]
[math] frac( \\log_3(2) + 1/2 \\log_3(2^8) )(1/3 \\log_3(2^3)) [/math]
[math] frac( \\log_3(2) + \\log_3((2^8)^{1/2}) )(\\log_3((2^3)^{1/3})) [/math]
[math] frac( \\log_3(2) + \\log_3(2^4) )(\\log_3(2)) [/math]
sapendo che:
[math] \\log_a (b_1 \cdot b_2) = \\log_a(b_1) + \\log_a (b_2)[/math]
possiamo scrivere
[math] frac( \\log_3(2^4 \cdot 2) )(\\log_3(2)) = frac( \\log_3(2^5) )(\\log_3(2)) [/math]
[math] frac( 5 \\log_3(2) )(\\log_3(2)) = 5 [/math]
