145th Problem 2017

Solution 1.

$\dfrac{1}{125}=25^{x+5}$

$\dfrac{1}{5^3}=5^{2(x+5)}$

$5^{-3}=5^{2x+10}$

$-3=2x+10$

$2x=-13$

$x=\dfrac{-13}{2}$

$x=-6.5$ $\boxed{\large\color{#3D99F6}answer}$

Solution 2.

Take the natural logarithm of both sides

$\ln~({\dfrac{1}{125})}=\ln~25^{x+5}$

$\ln~({\dfrac{1}{125})}=(x+5)~\ln~25$

$\dfrac{\ln~\frac{1}{125}}{\ln~25}=x+5$

$-1.5=x+5$

$x=-6.5$ $\boxed{\large\color{#3D99F6}answer}$

$\displaystyle \frac{1}{125} = 25^{x \ + \ 5}$

$\displaystyle 5^{- 3} = (5^2)^{x \ + \ 5} = 5^{2(x \ + \ 5)} = 5^{2x \ + \ 10}$

$\displaystyle - 3 = 2x + 10$

$\displaystyle x = - \frac{13}{2}$