Question

If $x=(100{)}^a,y=(10000{)}^b$ and $z=(10{)}^c$, express $\log \frac{10\sqrt{y}}{x^2{z}^3}$ in terms of $a,b,c$

Answer 1

$x=(100{)}^a$

Take log both sides,

$\log x=\log {100}^a$

$\log x=a\log 100$

$\log x=a\log {10}^2$

$\log x=2a$ .................. (1)

$y=(10000{)}^b$

$\log y=\log {10000}^b$

$\log y=b\log 10000$

$\log y=b\log {10}^4$

$\log y=4b$ ................ (2)

$z=(10{)}^c$

$\log z=\log {10}^c$

$\log z=c\log 10$

$\log z=c$ .................... (3)

$\log \frac{10\sqrt{y}}{x^2{z}^3}$$=\log 10\sqrt{y}-\log x^2{z}^3$

$=\log 10+\log \sqrt{y}-\log x^2-\log z^3$

$=1+\frac{1}{2}\log y-2\log x-3\log z$

$\log \frac{10\sqrt{y}}{x^2{z}^3}$$=1+2b-4a-3c$