Respuesta :
Answer:
[tex]=-8[/tex]
Step-by-step explanation:
[tex]\log _{0.5}\left(256\right)=x\\Switch\:sides\\x=\log _{0.5}\left(256\right)\\\mathrm{Rewrite\:}256\mathrm{\:in\:power-base\:form:}\quad 256=2^8\\x=\log _{0.5}\left(2^8\right)\\\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)\\\log _{0.5}\left(2^8\right)=8\log _{0.5}\left(2\right)\\x=8\log _{0.5}\left(2\right)\\0.5=2^{-1}\\x=8\log _{2^{-1}}\left(2\right)\\[/tex]
[tex]\mathrm{Apply\:log\:rule\:}\log _{a^b}\left(x\right)=\frac{1}{b}\log _a\left(x\right),\:\quad \mathrm{\:assuming\:}a\:\ge \:0\\x=8\cdot \frac{1}{-1}\log _2\left(2\right)\\\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1\\x=8\cdot \frac{1}{-1}\\\mathrm{Simplify}\\8\cdot \frac{1}{-1}\\\frac{1}{-1}=-1\\\frac{1}{-1}\\\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{-b}=-\frac{a}{b}\\=-\frac{1}{1}\\\mathrm{Apply\:rule}\:\frac{a}{1}=a\\=-1\\=8\left(-1\right)\\[/tex]
[tex]\mathrm{Remove\:parentheses}:\quad \left(-a\right)=-a\\=-8\cdot \:1\\\mathrm{Multiply\:the\:numbers:}\:8\cdot \:1=8\\=-8\\[/tex]
