Question

Desenvolva aplicando as propriedades dos logaritmos;

Se possível não coloque apenas a resposta;
S2

Answer 1

Prt.1/2

Resposta:

a)$\log _5\left(3\cdot \:4\right)=\log _5\left(3\right)+2\log _5\left(2\right)\quad \left(\mathrm{Decimal:\quad }\:1.54395\dots \right)$

b)$\log _4\left(2\cdot \:3\cdot \:5\right)=\frac{1}{2}\left(1+\log _2\left(15\right)\right)\quad \left(\mathrm{Decimal:\quad }\:2.45344\dots \right)$

c)$\log _5\left(\frac{2}{3}\right)=\log _5\left(2\right)-\log _5\left(3\right)\quad \left(\mathrm{Decimal:\quad }\:-0.25192\dots \right)$

d)$\log _{10}\left(\frac{2\cdot \:3}{5}\right)=\log _{10}\left(6\right)-\log _{10}\left(5\right)\quad \left(\mathrm{Decimal:\quad }\:0.07918\dots \right)$

e)$\log _3\left(\frac{a^3b^2}{c^4}\right)=3\log _3\left(a\right)+2\log _3\left(b\right)-4\log _3\left(c\right)$

f)$\log _{10}\left(\frac{a^3}{b^2\sqrt{c}}\right)=3\log _{10}\left(a\right)-2\log _{10}\left(b\right)-\log _{10}\left(\sqrt{c}\right)$

Explicação passo-a-passo:

a)

$\log _5\left(3\cdot \:4\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _a\left(xy\right)=\log _a\left(x\right)+\log _a\left(y\right)\\\\\log _5\left(3\cdot \:4\right)=\log _5\left(3\right)+\log _5\left(4\right)\\\\=\log _5\left(3\right)+\log _5\left(4\right)$

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$\log _5\left(4\right)\\\\\mathrm{Fatorar\:o\:numero:\:}\:4=2^2\\\\=\log _5\left(2^2\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right),\:\quad \:x>0\\\\\log _5\left(2^2\right)=2\log _5\left(2\right)\\\\=2\log _5\left(2\right)$

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$=\log _5\left(3\right)+2\log _5\left(2\right)$

b)

$\log _4\left(2\cdot \:3\cdot \:5\right)\\\\\mathrm{Fatorar\:o\:numero:\:}\:4=2^2\\\\=\log _{2^2}\left(2\cdot \:3\cdot \:5\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _{a^b}\left(x\right)=\frac{1}{b}\log _a\left(x\right),\:\quad \:a>0\\\\\log _{2^2}\left(2\cdot \:3\cdot \:5\right)=\frac{1}{2}\log _2\left(2\cdot \:3\cdot \:5\right)$

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$\log _2\left(2\cdot \:3\cdot \:5\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _a\left(xy\right)=\log _a\left(x\right)+\log _a\left(y\right)\\\\\log _2\left(2\cdot \:3\cdot \:5\right)=\log _2\left(2\right)+\log _2\left(3\right)+\log _2\left(5\right)\\\\=\log _2\left(2\right)+\log _2\left(3\right)+\log _2\left(5\right)$

$\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _a\left(a\right)=1\\\\\log _2\left(2\right)=1\\\\=1+\log _2\left(3\right)+\log _2\left(5\right)$

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$=\frac{1}{2}\left(1+\log _2\left(3\right)+\log _2\left(5\right)\right)$

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$1+\log _2\left(3\right)+\log _2\left(5\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _c\left(a\right)+\log _c\left(b\right)=\log _c\left(ab\right)\\\\\log _2\left(3\right)+\log _2\left(5\right)=\log _2\left(3\cdot \:5\right)\\\\=1+\log _2\left(3\cdot \:5\right)\\\\\mathrm{Multiplicar\:os\:numeros:}\:3\cdot \:5=15\\\\=1+\log _2\left(15\right)$

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$=\frac{1}{2}\left(1+\log _2\left(15\right)\right)$

c)

$\log _5\left(\frac{2}{3}\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _a\left(\frac{x}{y}\right)=\log _a\left(x\right)\:-\:\log _a\left(y\right)\\\\\log _5\left(\frac{2}{3}\right)=\log _5\left(2\right)-\log _5\left(3\right)\\\\=\log _5\left(2\right)-\log _5\left(3\right)$

d)

$\log _{10}\left(\frac{2\cdot \:3}{5}\right)\\\\\mathrm{Multiplicar\:os\:numeros:}\:2\cdot \:3=6\\\\=\log _{10}\left(\frac{6}{5}\right)\\\\\mathrm{Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad \log _a\left(\frac{x}{y}\right)=\log _a\left(x\right)\:-\:\log _a\left(y\right)\\\\\log _{10}\left(\frac{6}{5}\right)=\log _{10}\left(6\right)-\log _{10}\left(5\right)\\\\=\log _{10}\left(6\right)-\log _{10}\left(5\right)$

Prt 2 - <https://brainly.com.br/tarefa/32825146>