Hello!
Applying the base 10 logarithmic properties, we have:
[tex]\log _c\left(a\right)+\log _c\left(b\right)=\log _c\left(ab\right)[/tex]
[tex]\log _{10}\left(\sqrt{35} )+\log _{10}\left(\sqrt{2})=\log _{10}\left(\sqrt{35}*\sqrt{2})[/tex]
If we have:
[tex]Log\sqrt{35}\:+\:log\sqrt{2}-log\sqrt{7}[/tex]
So:
[tex]\log _{10}\left(\sqrt{35}\sqrt{2}\right)-\log _{10}\left(\sqrt{7}\right) =\:?[/tex]
[tex]\log _{10}\left(\sqrt{70}\right)-\log _{10}\left(\sqrt{7}\right) =\:?[/tex]
Applying the base 10 logarithmic properties, we have:
[tex]\log _c\left(a\right)-\log _c\left(b\right)=\log _c\left(\dfrac{a}{b}\right)[/tex]
[tex]\log _{10}\left(\sqrt{70} \right)-\log _{10}\left(\sqrt{7} \right)=\log _{10}\left(\dfrac{\sqrt{70}}{\sqrt{7}}\right)[/tex]
[tex]\log _{10}\left(\dfrac{\sqrt{70}}{\sqrt{7}}\right) =\:?[/tex]
[tex]= \dfrac{\sqrt{70}}{\sqrt{7}}[/tex]
[tex]= \dfrac{\sqrt{2}*\sqrt{5}*\sqrt{7}\!\!\!\!\!\!\!\dfrac{\hspace{0.4cm}}{~}}{\sqrt{7}\!\!\!\!\!\!\!\dfrac{\hspace{0.4cm}}{~}}[/tex]
[tex]= \sqrt{2}*\sqrt{5}[/tex]
[tex]= \sqrt{10}[/tex]
So:
[tex]= \log _{10}\left(\sqrt{10}\right)[/tex]
[tex]=\log _{10}\left(10^{\frac{1}{2}}\right)[/tex]
Applying the base 10 logarithmic properties, we have:
[tex]\log _a\left(x^b\right)=b* \log _a\left(x\right)[/tex]
[tex]\log _{10}\left({10}^\frac{1}{2} \right)=\dfrac{1}{2} * \log _{10}\left(10\right)[/tex]
[tex]= \dfrac{1}{2} * 1[/tex]
[tex]= \boxed{\frac{1}{2}}\:\:or\:\:\boxed{\:0.5}\end{array}}\qquad\checkmark[/tex]
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I Hope this helps, greetings ... DexteR! =)
