3 Marks Question

Question 13 Marks

Find value of $\log 0.0625$ to the base 2 .

Answer

$\log _2 0.0625=\log _2 \frac{625}{10000}$
$\begin{array}{l}=\log _2 \frac{5^4}{10^4} \\ =\log _2\left(\frac{5}{10}\right)^4 \\ =4 \log _2\left(\frac{1}{2}\right)\end{array}$
[Applying rule $\log _a m^n=n \log _a m$ ]
$=4\left[\log _2 1-\log _2 2\right]$
[Applying rule $\log _a\left(\frac{m}{n}\right)=\log _a m-\log _a{ }^n$ ]
$=4[0-1]$
[Applying rule $\log _a 1=0$ and $\log _a a=1$ ]
$=-4$

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Question 23 Marks

Find the value of $(125)^{2 / 3} \times \sqrt{25} \times \sqrt[3]{5^3} \times 5^{1 / 2}$.

Answer

$(125)^{2 / 3} \times \sqrt{25} \times \sqrt[3]{5^3} \times 5^{1 / 2}$
$=\left(5^3\right)^{2 / 3} \times 5 \times\left(5^3\right)^{1 / 3} \times 5^{1 / 2}$ [Using $\sqrt[r]{a}=(a)^{1 / r]}$
$=5^{\frac{3 \times 2}{3}} \times 5 \times 5^{\frac{3 \times 1}{3}} \times 5^{1 / 2}$ $\left[\operatorname{Using}\left(a^{m I}\right)^n=a^{m m}\right]$
$\begin{array}{l}=5^2 \times 5 \times 5 \times 5^{1 / 2} \\ =5^{2+1+1+\frac{1}{2}} \\ =5^{4+\frac{1}{2}}=5^{9 / 2}\end{array}$
$=5^{9 / 2}$ $\left[ U \operatorname{sing} a^m \cdot a^n=a^{m+n}\right]$

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Applied Maths 3 Marks Question

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