Question

Sendo log5 = a e log 3 = b, calcule, em função de a e b:
a)log 45
b) log 2
c) log de 15 na base 9
d) log de 50 na base 15

Answer 1

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$\boxed{\begin{array}{l}\tt a)~\sf \ell og45=\ell og(3^2\cdot5)\\\sf=2\ell og3+\ell og5=2b+a\\\tt b)~\sf\ell og2=\ell og\bigg(\dfrac{10}{5}\bigg)=\ell og10-\ell og5\\\sf=1-a\\\tt c)~\sf \ell og_915=\dfrac{\ell og15}{\ell og9}\\\sf=\dfrac{\ell og(3\cdot5)}{\ell og3^2}=\dfrac{\ell og3+\ell og5}{2\ell og3}=\dfrac{b+a}{2b}\\\\\tt d)~\sf\ell og_{15}50=\dfrac{\ell og50}{\ell og15}=\dfrac{\ell og(5\cdot10)}{\ell og(3\cdot5)}\\\sf=\dfrac{\ell og5+\ell og10}{\ell og3+\ell og5}\\\sf=\dfrac{a+1}{b+a}\end{array}}$