Szczegółowe wyjaśnienie:
Jeśli jest jakiś logarytm [tex]log_{a} b=c[/tex] to [tex]a^{c} =b[/tex]
1.
a) [tex]log_{3} 81=c[/tex] [tex]log_{3} 81=4[/tex]
[tex]3^{c} =81[/tex]
[tex]3^{c} =3^{4}[/tex]
[tex]c=4[/tex]
b) [tex]log_{2} \frac{1}{32} =c[/tex] [tex]log_{2} \frac{1}{32} =-5[/tex]
[tex]2^{c} =\frac{1}{32}[/tex]
[tex]2^{c} =32^{-1}[/tex]
[tex]2^{c} =(2^{5})^{-1}[/tex]
[tex]2^{c} =2^{-5}[/tex]
[tex]c=-5[/tex]
c) [tex]log_{5} \sqrt{125} =c[/tex] [tex]log_{5} \sqrt{125} =\frac{3}{2}[/tex]
[tex]5^{c} =\sqrt{125}[/tex]
[tex]5^{c} =125^{\frac{1}{2}[/tex]
[tex]5^{c} =(5^{3} )^{\frac{1}{2}}[/tex]
[tex]5^{c} =5^{\frac{3}{2} }[/tex]
[tex]c=\frac{3}{2}[/tex]
d) [tex]log_{3} \frac{1}{\sqrt{27} } =c[/tex] [tex]log_{3} \frac{1}{\sqrt{27} } =-\frac{3}{2}[/tex]
[tex]3^{c} =\frac{1}{\sqrt{27} }[/tex]
[tex]3^{c} =(\sqrt{27} )^{-1}[/tex]
[tex]3^{c} =27 ^{-\frac{1}{2} }[/tex]
[tex]3^{c} =(3^{3} )^{-\frac{1}{2} }[/tex]
[tex]3^{c} =3^{-\frac{3}{2} }[/tex]
[tex]c=-\frac{3}{2}[/tex]
e) [tex]log_{2} \sqrt{32} =c[/tex] [tex]log_{2} \sqrt{32} =\frac{5}{2}[/tex]
[tex]2^{c} =\sqrt{32}[/tex]
[tex]2^{c} =32^{\frac{1}{2} }[/tex]
[tex]2^{c} =(2^{5} )^{\frac{1}{2} }[/tex]
[tex]2^{c} =2^{\frac{5}{2} }[/tex]
[tex]c=\frac{5}{2}[/tex]
f) [tex]log_{2} \frac{1}{16} =c[/tex] [tex]log_{2} \frac{1}{16}=-4[/tex]
[tex]2^{c} =\frac{1}{16}[/tex]
[tex]2^{c} =16^{-1}[/tex]
[tex]2^{c} =(2^{4} )^{-1}[/tex]
[tex]2^{c} =2^{-4}[/tex]
[tex]c=-4[/tex]
