Question

Calcule os logaritmos a seguir.

a) log3 (raiz quadrada de 3)

b) log1/2 32

c) log0,25 0,5

d) log4 2(raiz quadrada de 2)

e) log0,00001

f) log25 125

Answer 1

Explicação passo-a-passo:

a)

$\sf log_{3}~\sqrt{3}=x$

$\sf 3^x=\sqrt{3}$

$\sf 3^x=3^{\frac{1}{2}}$

Igualando os expoentes:

$\sf \red{x=\dfrac{1}{2}}$

b)

$\sf log_{\frac{1}{2}}~32=x$

$\sf \Big(\dfrac{1}{2}\Big)^x=32$

$\sf (2^{-1})^x=2^5$

$\sf 2^{-x}=2^5$

Igualando os expoentes:

$\sf -x=5~~~\cdot(-1)$

$\sf \red{x=-5}$

c)

$\sf log_{0,25}~0,5=x$

$\sf 0,25^x=0,5$

$\sf \Big(\dfrac{25}{100}\Big)^x=\dfrac{5}{10}$

$\sf \Big(\dfrac{1}{4}\Big)^x=\dfrac{1}{2}$

$\sf \Big(\dfrac{1}{2^2}\Big)^x=\dfrac{1}{2}$

$\sf (2^{-2})^x=2^{-1}$

$\sf 2^{-2x}=2^{-1}$

Igualando os expoentes:

$\sf -2x=-1~~~\cdot(-1)$

$\sf 2x=1$

$\sf \red{x=\dfrac{1}{2}}$

d)

$\sf log_{4}~2\sqrt{2}=x$

$\sf 4^x=2\sqrt{2}$

$\sf (2^2)^x=\sqrt{2^2\cdot2}$

$\sf 2^{2x}=\sqrt{2^3}$

$\sf 2^{2x}=2^{\frac{3}{2}}$

Igualando os expoentes:

$\sf 2x=\dfrac{3}{2}$

$\sf 2\cdot2x=3$

$\sf 4x=3$

$\sf \red{x=\dfrac{3}{4}}$

e)

$\sf log~0,00001=x$

$\sf 10^x=0,00001$

$\sf 10^x=\dfrac{1}{100000}$

$\sf 10^x=\dfrac{1}{10^5}$

$\sf 10^x=10^{-5}$

Igualando os expoentes:

$\sf \red{x=-5}$

f)

$\sf log_{25}~125=x$

$\sf 25^x=125$

$\sf (5^2)^x=5^3$

$\sf 5^{2x}=3$

Igualando os expoentes:

$\sf 2x=3$

$\sf \red{x=\dfrac{3}{2}}$