Question

If log2 to the base 10=a and log3 to the base 10=b,then express each of the following in terms of 'a'&'b'.
(A)log 5.4 to the base 10
(B)log 25/8 to the base 10

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{are}\mathrm{given}\mathrm{that} \\ {\log }_{10}2=\mathrm{a}\mathrm{and}{\log }_{10}3=\mathrm{b} \\ \left(\mathrm{A}\right) \\ {\log }_{10}5.4={\log }_{10}\frac{54}{10}={\log }_{10}54-{\log }_{10}10\left(\mathrm{As},\log \left(\frac{\mathrm{m}}{\mathrm{n}}\right)=\log \mathrm{m}-\log \mathrm{n}\right) \\ ={\log }_{10}\left(2\times 3^3\right)-1\left(\mathrm{As},{\log }_{10}10=1\right) \\ ={\log }_{10}2-{\log }_{10}{3}^3-1\left(\mathrm{As},\log \left(\mathrm{m}\times \mathrm{n}\right)=\log \mathrm{m}+\log \mathrm{n}\right) \\ ={\log }_{10}2-3{\log }_{10}3-1\left(\mathrm{As},\log {\mathrm{m}}^{\mathrm{n}}=\mathrm{n}\log \mathrm{m}\right) \\ =\mathrm{a}-3\mathrm{b}-1 \\ \left(\mathrm{B}\right) \\ \mathrm{similarly},{\log }_{10}\frac{25}{8}\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as} \\ {\log }_{10}\left(\frac{25}{8}\right)={\log }_{10}25-{\log }_{10}8\left(\mathrm{As},\log \left(\frac{\mathrm{m}}{\mathrm{n}}\right)=\log \mathrm{m}-\log \mathrm{n}\right) \\ ={\log }_{10}{5}^2-{\log }_{10}{2}^3 \\ =2{\log }_{10}5-3{\log }_{10}2\left(\mathrm{As},\log {\mathrm{m}}^{\mathrm{n}}=\mathrm{n}\log \mathrm{m}\right) \\ =2{\log }_{10}\frac{10}{2}-3{\log }_{10}2 \\ =2{\log }_{10}10-2{\log }_{10}2-3{\log }_{10}2\left(\mathrm{As},\log \left(\frac{\mathrm{m}}{\mathrm{n}}\right)=\log \mathrm{m}-\log \mathrm{n}\right) \\ =2-5{\log }_{10}2=2-5\mathrm{a}. \end{gathered}$$