Question

Determine o valor de:

Answer 1

a)
$S=\log _2\left(16\right)+\log _9\left(\sqrt[3]{3}\right)-\log _2\left(0,25\right)$

$\log _2\left(16\right)=x\rightarrow \:16=2^x\rightarrow \:2^4=2^x\rightarrow \:x=4$

$\log _9\left(\sqrt[3]{3}\right)=y\:\rightarrow \:\:\log _9\:3^{\frac{1}{3}}=y\rightarrow \:\frac{1}{3}\log _9\left(3\right)=y\rightarrow \:\frac{1}{3}\cdot \frac{1}{2}=y= \frac{1}{6}$

$\log _8\left(0,25\right)=y\rightarrow \:\frac{\log _2\left(0,25\right)}{\log 2\left(8\right)}=y\rightarrow \:\frac{-2}{3}=y=-\frac{2}{3}$

⇒$4+ \frac{1}{6}-( -\frac{2}{3} )\rightarrow4+\frac{1}{\:6}+\frac{2}{3}= \frac{29}{6}$

b)
$a^{\log _a\left(b\right)}=b$

$2^{^{\log _2\left(15\right)}}=15$

c)
$25^{\log _5\left(4\right)\cdot \log _{16}\left(7\right)}$

$(5^2)^{\log _5\left(4\right)\cdot \log _{16}\left(7\right)}$

$5^{\log _5\left(4\right)\cdot \log _{16}\left(7\right)\cdot2}$

$4^ \log _{16}\left(7\right)\cdot2}$

$(4^2)^ \log _{16}\left(7\right)}$

$16^ \log _{16}\left(7\right)}=7$

d)

e)
$-\log _2\left(\frac{1024}{\sqrt[3]{256}}\right)$

$-\log _2\left(\frac{2^{10}}{\sqrt[3]{2^{8}}}\right)$

$-\log _2\left(\frac{2^{10}}{{2^{ \frac{1}{3}\cdot 8}}}\right)$

$-\log _2\left(2^{10}\cdot2^{-\frac{8}{3} }\right)$

$-\log _2\left(2^{10-\frac{8}{3}\right)$

$-\log _2\left(2^\frac{22}{3}\right)$

$\log _2\left(2^\frac{22}{3}\right)^{-1}\rightarrow\log _2\left(2^{\frac{22}{2}\cdot \left(-1\right)}\right)\rightarrow\log_2 (2^{-\frac{22}{3}})=y\rightarrow2^y=2^{-\frac{22}{3}}$

$y= -\frac{22}{3}$