Exponents and Logs – Question bank (Qn 1 -5)

\begin{aligned}<br /> \frac{3^{x+2}+3^x}{3^x-3^{x+1}} & =\frac{3^x \cdot 3^2+3^x}{3^x-3^x \cdot 3^1} \\<br /> & =\frac{3^x[9+1]}{3^x[1-3]} \\<br /> & =\frac{10}{-2} \\<br /> & =-5<br /> \end{aligned}

[a]  \log 20

\begin{aligned}<br /> \log 20 & =\log (5 \times 4) \\<br /> & =\log 5+\log 4 \\<br /> \log 20 & =p+q<br /> \end{aligned}

[b]  \log 16

 \begin{aligned}<br /> \log 16 & =\log 4^2 \\<br /> & =2 \log 4 \\<br /> & =2 q<br /> \end{aligned}

[c]  \log 16

\begin{aligned}<br /> \log 80 & =\log (5 \times 16) \\<br /> & =\log 5+\log 16 \\<br /> \log 20 & =p+2q<br /> \end{aligned}

\begin{aligned}<br /> \log _3 5+\log _9 4 & =\log _3 5+\frac{\log _3 4}{\log _3 9} \\<br /> & =\frac{\log _3 5}{1}+\frac{\log _3 4}{2} \\<br /> & =\frac{2 \log _3 5+\log _3 4}{2} \\<br /> & =\frac{\log _3 5^2+\log _3 4}{2} \\<br /> & =\frac{\log _3 25+\log _3 4}{2} \\<br /> & =\frac{\log _3(25 \times 4)}{2} \\<br /> & =\frac{\log _3 100}{2} \\<br /> & =\frac{1}{2} \log _3 100 \\<br /> & =\log _3(100)^{1 / 2} \\<br /> & =\log _3 \sqrt{100} \\<br /> & =\log _3 10<br /> \end{aligned}

\left(\frac{1}{2} \cdot \log _2 a\right)+\left(3 \cdot \log _2 b\right)-\left(\frac{3}{2} \cdot \log _2 c\right)=\log _2 a^{\frac{1}{2}}+\log _2 b^3+\left(-\frac{3}{2} \cdot \log _2 c\right)

=\log _2 \sqrt{a}+\log _2 b^3+\log _2 c^{\left(-\frac{3}{2}\right)}

=\log _2 \sqrt{a}+\log _2 b^3+\log _2\left(\frac{1}{\sqrt{c^3}}\right)

=\log _2\left(\sqrt{a} \cdot b^3 \cdot \frac{1}{\sqrt{c^3}}\right)

=\log _2\left(\frac{\sqrt{a} \cdot b^3}{\sqrt{c^3}}\right)

[a]  \log \frac{x}{\sqrt{y z}}. 

\begin{aligned}<br /> \log \frac{x}{\sqrt{y z}} & =\log x-\log \sqrt{2} \\<br /> & =\log x-\frac{1}{2}[\log y+\log 2] \\<br /> & =a-\frac{1}{2}(b+c)<br /> \end{aligned}

[b]  \log \sqrt[3]{\frac{x y^2}{z^3}}. 

\begin{aligned}<br /> \log \frac{x^2 y^3}{2} & =\log x^2+\log y^3-\log z \\<br /> & =2 \log x+3 \log y-\log z \\<br /> & =2 a+3 b-c<br /> \end{aligned}

[c]  \log \frac{x^2 y^3}{2}

\begin{aligned}<br /> \log \sqrt[3]{\frac{x y^2}{z^3}} & =\log \left(\frac{x y^2}{z^3}\right)^{1 / 3} \\<br /> & =\frac{1}{3} \log \frac{x y^2}{z^3} \\<br /> & =\frac{1}{3}\left[\log x+\log y^2-\log z^3\right] \\<br /> & =\frac{1}{3}[\log x+2 \log y-3 \log z] \\<br /> & =\frac{1}{3}[a+2 b-3 c]<br /> \end{aligned} 

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