Question

Considerando log2 = 0,30, log3 = 0,48 log 5= 0, 70, determine:
Log5^30=
Log9^2/5=
Log6^90=
Log15^√16=

Answer 1

Explicação passo-a-passo:

a)

$\sf log_{5}~30=\dfrac{log~30}{log~5}$

$\sf log_{5}~30=\dfrac{log~(2\cdot3\cdot5)}{log~5}$

$\sf log_{5}~30=\dfrac{log~2+log~3+log~5}{log~5}$

$\sf log_{5}~30=\dfrac{0,30+0,48+0,70}{0,70}$

$\sf log_{5}~30=\dfrac{1,48}{0,70}$

$\sf log_{5}~30=2,11$

b)

$\sf log_{9}~\dfrac{2}{5}=\dfrac{log~\frac{2}{5}}{log~9}$

$\sf log_{9}~\dfrac{2}{5}=\dfrac{log~\frac{2}{5}}{log~3^2}$

$\sf log_{9}~\dfrac{2}{5}=\dfrac{log~2-log~5}{2\cdot log~3}$

$\sf log_{9}~\dfrac{2}{5}=\dfrac{0,30-0,70}{2\cdot0,48}$

$\sf log_{9}~\dfrac{2}{5}=\dfrac{-0,40}{0,96}$

$\sf log_{9}~\dfrac{2}{5}=-0,417$

c)

$\sf log_{6}~90=\dfrac{log~90}{log~6}$

$\sf log_{6}~90=\dfrac{log~(2\cdot3^2\cdot5)}{log~(2\cdot3)}$

$\sf log_{6}~90=\dfrac{log~2+log~3^2+log~5}{log~2+log~3}$

$\sf log_{6}~90=\dfrac{log~2+2\cdot log~3+log~5}{log~2+log~3}$

$\sf log_{6}~90=\dfrac{0,30+2\cdot0,48+0,70}{0,30+0,48}$

$\sf log_{6}~90=\dfrac{0,30+0,96+0,70}{0,30+0,48}$

$\sf log_{6}~90=\dfrac{1,96}{0,78}$

$\sf log_{6}~90=2,51$

d)

$\sf log_{15}~\sqrt{16}=\dfrac{log~\sqrt{16}}{log~15}$

$\sf log_{15}~\sqrt{16}=\dfrac{log~4}{log~(3\cdot5)}$

$\sf log_{15}~\sqrt{16}=\dfrac{log~2^2}{log~(3\cdot5)}$

$\sf log_{15}~\sqrt{16}=\dfrac{2\cdot log~2}{log~3+log~5}$

$\sf log_{15}~\sqrt{16}=\dfrac{2\cdot0,30}{0,48+0,70}$

$\sf log_{15}~\sqrt{16}=\dfrac{0,60}{1,18}$

$\sf log_{15}~\sqrt{16}=0,508$