Question

Calcule o valor de: log de 3 na base 2 ×log de 5 na base 4× log2 na base 5× log de 4 na base 9

Answer 1

$log_{_2}3\,\times\,log_{_{4}}5\,\times\,log_{_5}2\,\times\,log_{_9}4\\\\\\Vamos\;utilizar\;a\;propriedade\;de\;mudanca\;de\;base:\;log_{_b}a=\frac{log_{_c}a}{log_{_c}b}\\\\\frac{log_{_{10}}3}{log_{_{10}}2}\,\times\,\frac{log_{_{10}}5}{log_{_{10}}4}\,\times\,\frac{log_{_{10}}2}{log_{_{10}}5}\,\times\,\frac{log_{_{10}}4}{log_{_{10}}9}\\\\Podemos\;simplificar\;alguns\;numeradores\;com\;denominadores:$

$\frac{log_{_{10}}3}{1}\,\times\,\frac{1}{1}\,\times\,\frac{1}{1}\,\times\,\frac{1}{log_{_{10}}9}\\\\\\\frac{log_{_{10}}3}{log_{_{10}}9}\\\\\\Utilizando\; novamente\; a\; troca\; de \;base:\\\\\frac{log_{_{10}}3}{log_{_{10}}9}=log_{_9}3$

Podemos resolver $log_{_9}3$ aplicando a definicao de log:

$log_{_9}3=x\\\\\\3=9^x\\\\\\3^1=(3^2)^x\\\\\\3^1=3^{2x}\\\\\\2x=1\\\\\\x=\frac{1}{2}$

Resp: O valor da expressão é 1/2 ou 0,5

Answer 2

Explicação passo-a-passo:

$\log_23\times\log_45\times\log_52\times\log_94=\\ \\ mudar~~para~~base~~2\\ \\\log_23\times{\log_25 \over\log_24}\times{\log_22\over\log_25}\times{\log_24\over\log_29}=\\ \\ cancelar~~~\log_25~~com~~\log_25~~~e~~~\log_24~~~com~~\log_24$

$como~~\log_22=1\\ \\ fica\\ \\ \log_23\times{1\over\log_29}=\\ \\ {\log_23\over\log_23^2}={\log_23\over2\log_23}={1\over2}~~ ou~~ 0,5\\ \\ cancela~~\log_23$