Question

Write the following logarithmic form:
i) $\rm E = \frac{1}{2} mv^{2}$
ii) $32^{\frac{3}{5}} = 8$

Answer 1

To convert these expression in logarithmic form,taking log both sides

$1) \rm E = \frac{1}{2} mv^{2} \\ \\ log \: E = log(\frac{1}{2} mv^{2}) \\ \\ = log( \frac{m {v}^{2} }{2} ) \\ \\ = log(m {v}^{2} ) - log \: 2 \\ \\ = log \: m + log \: {v}^{2} - log \: 2 \\ \\ log \: E = log \: m + 2log \: {v} - log \: 2$
$2) 32^{\frac{3}{5}} = 8 \\ \\ log \: 32^{\frac{3}{5}} = log \: 8 \\ \\ \\ log \: ( {2}^{5})^{\frac{3}{5}} = log \: {2}^{3} \\ \\log \: {2}^{3} = 3log \: 2 \\ \\ 3log \: 2 = 3 \: log \: 2 \\ \\ or \\ \\ \frac{3}{5} log \: 32 = log \: 8$
Hope it helps you.