Question

Calcule o valor da expressão : [ log2 64 + log6 36 ] - log7  1/49.

Answer 1

$\textsf{Resolva por partes:}\\\\\\\mathsf{log_{2}64}\\\\\mathsf{log_2 64=x}\\\\\mathsf{2^x=64~~~~(fatore~64~para~igualar~as~bases)}\\\\\mathsf{2^x=2^6~~~~(iguale~os~expoentes)}\\\\\fbox{ \mathsf{x=6} }$


$\mathsf{log_6 36}\\\\\mathsf{log_6 36=y}\\\\\mathsf{6^y=36~~~~(fatore~36~para~igualar~as~bases)}\\\\\mathsf{6^y=6^2~~~~(iguale~os~expoentes)}\\\\\fbox{ \mathsf{y=2} }$


$\mathsf{log_7\dfrac{1}{49}}\\\\\\\mathsf{log_7\dfrac{1}{49}=z}\\\\\\\mathsf{7^z=\dfrac{1}{49}~~~~(reescreva~a~fra\c{c}\~ao~de~outra~forma)}\\\\\\\mathsf{7^z=49^{-1}~~~~(fatore~49^{-1}~para~igualar~as~bases)}\\\\\\\mathsf{7^z=(7^2)^{-1}~~~~(pela~propriedade~exponencial~(a^b)^c=a^{b\cdot c}...)}\\\\\\\mathsf{7^z=7^{-2}~~~~(iguale~os~expoentes)}\\\\\\\fbox{ \mathsf{z=-2} }$


$\mathsf{(log_264+log_636)-log_7\dfrac{1}{49}}\\\\\\\mathsf{=(x+y)-z}\\\\\\\mathsf{=(6+2)-(-2)}\\\\\\\mathsf{=8+2}\\\\\\\large\fbox{ \mathsf{=10} }$