`log_b(27) = 3` Solve for x or b

Friday, June 28, 2013

$\displaystyle{\log}_{{b}}{\left({27}\right)}={3}$

To solve, convert this to exponential equation.

Take note that if a logarithmic equation is in the form

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

its equivalent exponential equation is:

$\displaystyle{x}={{b}}^{{y}}$

So converting

$\displaystyle{\log}_{{b}}{\left({27}\right)}={3}$

to exponential equation, it becomes:

$\displaystyle{27}={{b}}^{{3}}$

Then, factor the left side.

$\displaystyle{{3}}^{{3}}={{b}}^{{3}}$

And isolate the b by taking the cube root of both sides.

$\displaystyle{\sqrt[{{3}}]{{{{3}}^{{3}}}}}={\sqrt[{{3}}]{{{{b}}^{{3}}}}}$

$\displaystyle{3}={b}$

Therefore, $\displaystyle{b}={3}$ .

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