4.1) Simplifique cada uma das seguintes designações:

(a) ${\log _a}x + {\log _a}(4x) - {\log _a}2$;	(b) $3{\log _a}\left( x \right) + 2{\log _a}\left( {3x} \right)$;	(c) $2\ln \left( {3x} \right) - {e^{\ln 3}} - \ln \left( x \right)$;
(d) ${2^{{{\log }_2}\left( x \right) + 3}} + 3\ln \left( {2x} \right) - 2\ln \left( x \right) - 8x$;	(e) ${\log _2}\frac{8}{{128}} - {\log _2}\frac{1}{{2x}}$	(f) ${2^{5 - 2{{\log }_2}\left( {8x} \right)}}$
(g) $\ln {e^{2t}} - \ln \frac{1}{t}$;	(h) ${\log _9}{3^{6x+12}}-6$;	(i) ${10^{{{\log }_{10}}t}}$
(j) ${e^{2\ln x - \frac{1}{2}}} \times \frac{{\sqrt e }}{{{x^2}}} - 1$;	(l) $\log_{\frac{1}{3}}{3^{4x}}$	(m) ${e^{\ln x + \ln 2}}$

Rui

4.2) Escreva como um só logaritmo:

(a) $2\log_{10}3 + \log_{10}5$;	(b) $2+\frac{1}{2}\log_{10}5$;	(c) $3\left(1-\log_{10}a\right)$;
(d) $3\left(\log_{10}x+\log_{10}y-\log_{10}z\right)$;	(e) $3+2\ln x$;	(f) $10+x+\log(5x)$.

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4.3) Mostre que:

(a) $\log_2\left(x-x^2\right)-\log_4x^2=\log_2\left(1-x\right)$, $\forall x\in ]0,1[$.

(b) $2\log_9(x+1)+3\log_3 x=\log_3\left(x^4+x^3\right)$, $\forall x\in \mathbb{R}^+$.

(c) $8\log_4\left(3x+1\right)^3-4\log_2(3x+1)=16\log_4(3x+1)$, $\forall x\in \left]-\frac{1}{3},+\infty\right[$.

           

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