How do you solve 25^x=125^(x-2)?

Jun 5, 2017

$x = 6$

Explanation:

In this exponential equation, the clue is that both $25 \mathmr{and} 125$ are powers of $5$.

Write them both with a base of $5$

${25}^{x} = {125}^{x - 2}$

${\left({5}^{2}\right)}^{x} = {\left({5}^{3}\right)}^{x - 2}$

Raising a power to a power $\rightarrow$multiply the indices:

${\textcolor{b l u e}{5}}^{\textcolor{red}{\left(2 x\right)}} = {\textcolor{b l u e}{5}}^{\textcolor{red}{\left(3 x - 6\right)}}$

In this equation the bases are both equal (to $5$,) so the indices also have to be equal.

$\therefore \textcolor{red}{2 x = 3 x - 6} \text{ } \leftarrow$ solve this linear equation for $x$

$6 = x$