log 2 x - log 4 x = 2 find x

Wednesday, February 20, 2013

log 2 x - log 4 x = 2 find x

The equation $\displaystyle{\log}_{{2}}{x}-{\log}_{{4}}{x}={2}$ has to be solved for
x.

First convert all the logarithm to a common base. As the
numbers here are 2 and powers to two we use the base 2. Use the following property
useful when changing the base of logarithms.

$\displaystyle{\left({\log}_{{b}}{x}\right)}=$
(log_n x)/(log_n b)

$\displaystyle{\log}_{{2}}{x}-{\log}_{{4}}{x}=$
2

$\displaystyle{\log}_{{2}}{x}-\frac{{{\log}_{{2}}{x}}}{{{\log}_{{2}}{4}}}=$
2

$\displaystyle{\log}_{{2}}{x}-\frac{{{\log}_{{2}}{x}}}{{{\log}_{{2}}{{2}}^{{2}}}}=$
2

use the property $\displaystyle{{\log{{a}}}}^{{b}}={b}\cdot{\log{}}$
a

$\displaystyle{\log}_{{2}}{x}-\frac{{{\log}_{{2}}{x}}}{{{2}\cdot{\log}_{{2}}{2}}}=$
2

Now $\displaystyle{\log}_{{b}}{b}={1}$

$\displaystyle{\log}_{{2}}{x}$
- (log_2 x)/(2*1) = 2

$\displaystyle{\left(\frac{{1}}{{2}}\right)}\cdot{\log}_{{2}}{x}={2}$

$\displaystyle{\log}_{{2}}{x}={4}$

If $\displaystyle{\log}_{{b}}{x}=$
y $\displaystyle,$ x = b^y

$\displaystyle{x}={{4}}^{{2}}$

x =
16

The required solution is x = 16

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Wednesday, February 20, 2013