Notes: `5^(x-4)=25^(x-6)` Solve the equation.

Friday, 25 October 2013

$\displaystyle{{5}}^{{{x}-{4}}}={{25}}^{{{x}-{6}}}$ Solve the equation.

$\displaystyle{{5}}^{{{x}-{4}}}={{25}}^{{{x}-{6}}}$

To solve, factor the 25.

$\displaystyle{{5}}^{{{x}-{4}}}={{\left({{5}}^{{2}}\right)}}^{{{x}-{6}}}$

To simplify the right side, apply the exponent rule $\displaystyle{{\left({{a}}^{{m}}\right)}}^{{n}}={{a}}^{{{m}\cdot{n}}}$ .

$\displaystyle{{5}}^{{{x}-{4}}}={{5}}^{{{2}\cdot{\left({x}-{6}\right)}}}$

$\displaystyle{{5}}^{{{x}-{4}}}={{5}}^{{{2}{x}-{12}}}$

Since both sides have the same base, to solve for the value of x, set the exponent at the left equal to the exponent at the right.

$\displaystyle{x}-{4}={2}{x}-{12}$

$\displaystyle{x}-{2}{x}={4}-{12}$

$\displaystyle-{x}=-{8}$

$\displaystyle{x}={8}$

Therefore, the solution is $\displaystyle{x}={8}$ .

$\displaystyle{{5}}^{{{x}-{4}}}={{25}}^{{{x}-{6}}}$

To solve, factor the 25.

$\displaystyle{{5}}^{{{x}-{4}}}={{\left({{5}}^{{2}}\right)}}^{{{x}-{6}}}$

To simplify the right side, apply the exponent rule $\displaystyle{{\left({{a}}^{{m}}\right)}}^{{n}}={{a}}^{{{m}\cdot{n}}}$ .

$\displaystyle{{5}}^{{{x}-{4}}}={{5}}^{{{2}\cdot{\left({x}-{6}\right)}}}$

$\displaystyle{{5}}^{{{x}-{4}}}={{5}}^{{{2}{x}-{12}}}$

Since both sides have the same base, to solve for the value of x, set the exponent at the left equal to the exponent at the right.

$\displaystyle{x}-{4}={2}{x}-{12}$

$\displaystyle{x}-{2}{x}={4}-{12}$

$\displaystyle-{x}=-{8}$

$\displaystyle{x}={8}$

Therefore, the solution is $\displaystyle{x}={8}$ .

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)