01.
log[1/2] (1/32)=x
2^(-5) =2^(-x)
x=5
log[y] 81 =4
3^4 = y^4
y=3
log (xy/45) / log 3
[log x + log y -log 45]/ log 3
[log x + log y -log (3²*5)]/ log 3
[log x + log y -log 3² - log 5]/ log 3
[log 5 + log 3 -2*log 3 - log 5]/ log 3
[ -*log 3]/ log 3
=-1
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02.
a*b*c=1
log[b] c * log[c] a * log[a] b =1
logc/log b * loga /log c * log b /log a =1
=log a * log b * log c /(log a * log b * log c )
= log b * log c /( log b * log c )
= log c /( log c )
= 1 ==> CQP
# cqc ==>como queríamos provar
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03.
log[x] a =6 ==> log a=6 * log x
log[x] b=4 ==> log b= 4 * log x
log[x] c =2 ==> log c = 2 * log x
log[c] (abc)^(1/2)
=(1/2) * [log a +log b+log c]
=(1/2) * [6 * log x +4 * log x+2 * log x]
=(1/2) * [12 * log x]
= 6 * log x
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04.
log[a] a³* b²=m
log a³* b² / log a =m
(log a³ + log b²)=m * log a
(3*log a + log b²)=m * log a
log a *(m-3) =2* log b
log a/log b = 2/(m-3)
log[b] = 2/(m-3) é a resposta
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05.
log (a^(-1))^(1/2) - log b²
log a^(-1/2) - 2 log b
(1/2)* log a - 2*log b
=(1/2) * 2 -2 * 3
=1 -6
= -5
Resposta:
Explicação passo a passo:
[tex]Quest\tilde{a} o\:1)[/tex]
[tex]log_{\:y} \:81=4[/tex]
[tex]y^{4}=81[/tex]
[tex]y=\sqrt[4]{81}[/tex]
[tex]y=\sqrt[4]{3^{4} }[/tex]
[tex]y=3[/tex]
[tex]log_{\:\frac{1}{2} } \:(\frac{1}{32}) =x[/tex]
[tex](\frac{1}{2}) ^{x} =\frac{1}{32}[/tex]
[tex](\frac{1}{2}) ^{x} =\frac{1}{2^{5} }[/tex]
[tex](\frac{1}{2}) ^{x} =(\frac{1}{2}) ^{5}[/tex]
[tex]x=5[/tex]
[tex]\frac{x\:.\:y}{45}=\frac{5\:.\:3}{45} =\frac{15}{45} =\frac{1}{3}[/tex]
[tex]Portanto...[/tex]
[tex]log_{\:3} \:(\frac{x\:.\:y}{45})=log_{\:3} \:(\frac{1}{3})=log_{\:3} \:(3^{-1}) =(-1)\:.\:log_{\:3} \:3=(-1)\:.\:1=-1[/tex]
[tex]Quest\tilde{a} o\:2)[/tex]
[tex]a\times\:b\times\:c=\:?[/tex]
[tex]Vamos\:passar\:todos\:para\:a\:base\;10\:=>(base\;invisivel)[/tex]
[tex]\frac{log\:c}{log\:b} \times\frac{log\:a}{log\:c}\times\:\frac{log\:b}{log\:a}=1[/tex]
[tex]Por\:que\:deu\:1?[/tex]
[tex]Resposta:\;Cancelando\:os\:termos\;iguais\:em\;cima\:e\:embaixo\:ok![/tex]
[tex]Quest\tilde{a} o\:3)[/tex]
[tex]log_{\:x} \:a=6=>a=x^{6}[/tex]
[tex]log_{\:x} \:b=4=>b=x^{4}[/tex]
[tex]log_{\:x} \:c=2=>c=x^{2}[/tex]
[tex]a.b.c=x^{6}.x^{4}.x^{2} =x^{12}[/tex]
[tex]\sqrt{abc}=(abc)^{\frac{1}{2} }[/tex]
[tex]log_{\:c} \:\sqrt{abc} =log_{\:c} \:(abc)^{\frac{1}{2} } =\frac{1}{2}\:.\:log_{\:c} \:(abc) =\frac{1}{2}\:.\:log_{\:c} \:(x^{12}) =\frac{1}{2}\:.\:12\:.\:log_{\:c}\:x=\\\\6\:.\:\frac{1}{log_{\:x}\:c } =6\:.\:\frac{1}{2}=3[/tex]
[tex]Quest\tilde{a} o\:4)[/tex]
[tex]log_{\:b}\:a=?[/tex]
[tex]log_{\:a} \:(a^{3}\:.\:b^{2} )=m[/tex]
[tex]log_{\:a} \:(a^{3}\:.\:b^{2} )=log_{\:a} \:a^{3} +log_{\:a}\:b^{2} =3\:.\:log_{\:a}\:a+2\:.\:log_{\:a}\:b=3\:.\:1+2\:.\:log_{\:a}\:b[/tex]
[tex]Portanto...[/tex]
[tex]3\:.\:1+2\:.\:log_{\:a}\:b=m[/tex]
[tex]3+2\:.\:log_{\:a}\:b=m[/tex]
[tex]2\:.\:log_{\:a}\:b=m-3[/tex]
[tex]Observacao...[/tex]
[tex]log_{\:a}\:b =\frac{1}{log_{\:b}\:a }[/tex]
[tex]2\:.\:\frac{1}{log_{\:b}\:a }=m-3[/tex]
[tex]\frac{2}{log_{\:b}\:a }=m-3[/tex]
[tex]log_{\:b}\:a\: \;.\:(m-3)=2[/tex]
[tex]log_{\:b}\:a=\frac{2}{(m-3)}[/tex]
[tex]Quest\tilde{a} o\:5)[/tex]
[tex]log\:a=2[/tex]
[tex]log\:b=3[/tex]
[tex]log(\:\frac{\sqrt{\frac{1}{a} } }{b^{2} }) =[/tex]
[tex]log\:\sqrt{\frac{1}{a} } \:\:-\:\:log\:b^{2} =[/tex]
[tex]log\:(\frac{1}{a}) ^{\frac{1}{2} } \:-\:2\:.\:log\:b=[/tex]
[tex]Observacao:[/tex]
[tex]\frac{1}{a} =a^{-1}[/tex]
[tex]\frac{1}{2}\:.\: log\:(a^{-1}) -2\:.\:log\:b=[/tex]
[tex]\frac{1}{2}\:.\:(-1)\:.\:log\:a-2\:.\:log\:b=[/tex]
[tex]-\frac{1}{2}\:.\:log\:a\:-\:2\:.\:log\:b=[/tex]
[tex]-\frac{1}{2}\:.\:(2)\:-\:2\:.\:(3)=[/tex]
[tex]-\frac{2}{2}\:-\:6=[/tex]
[tex]-1-6=[/tex]
[tex]-7[/tex]
[tex]Ufa![/tex]
[tex]Bye![/tex]
[tex]Jean318[/tex]
